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Preferential interaction networks: A dynamic model for brain synchronization networks
Journal Pre-proof Preferential interaction networks: A dynamic model for brain synchronization networks R.A. Sousa, V.N.A. Lula-Rocha, T. Toutain, R.S. Rosário, E.C.B. Cambui, J.G.V. Miranda
PII: DOI: Reference:
S0378-4371(20)30070-4 https://doi.org/10.1016/j.physa.2020.124259 PHYSA 124259
To appear in:
Physica A
Received date : 1 April 2019 Revised date : 30 December 2019 Please cite this article as: R.A. Sousa, V.N.A. Lula-Rocha, T. Toutain et al., Preferential interaction networks: A dynamic model for brain synchronization networks, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2020.124259. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
*Highlights (for review)
Journal Pre-proof Highlights:
of
p ro
Pr e-
al
urn
A network computational model testing the hypothesis of preferential interaction was created. The dependence of the model on the weight increment and network size and their limits were evaluated. The electroencephalogram (EEG) of four healthy subjects were collected to be used as model validation. The motif synchronization method was applied to the EEG data generating weighted synchronization networks. The network weight distribution, generated by the model, were compared with those generated by the EEG.
Jo
*Manuscript Click here to view linked References
Journal Pre-proof Preferential interaction networks: A dynamic model for brain synchronization networks 1
1
2
1
2
R.A. Sousa, V.N.A Lula-Rocha, T. Toutain, R.S. Rosário, E.C.B. Cambui, and J.G.V. Miranda 1 2
1,
Institute of Physics, Federal University of Bahia (IF-UFBA), 40210-340 Salvador, Brazil
Institute of Biology, Federal University of Bahia (IBIO-UFBA), 40,170-110 Salvador, Brazil
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of
In this work, we propose a weighted network model called a preferential interaction network (PIN), whose construction is based on mechanisms similar to those of the scalefree network model developed by Albert and Barabási. The PIN aims to reproduce the behaviour of real systems of fixed size in which stronger connections are reinforced over time. Its composition starts with a network with fixed nodes, and obeys two basic processes: firstly, an increase in the edge weights, and secondly, a preferential interaction is added. Preferential interaction is defined as the tendency whereby edges with larger weights are more likely to be chosen in a iteration. PIN has three parameters: the size of the network, the rate of increase of the weights, and the number of iterations. These parameters were used to fit the model to EEG brain synchronization networks of healthy subjects. The complementary cumulative distribution of weight in the model was compared to the corresponded distribution in EEG networks. The shape of the model distribution showed similar to those of the EEG data networks and the PIN model topology index are closer than a random null model, suggesting that a preferential interaction process represents the core of brain synchronization dynamics.
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Keywords: complex networks; preferential interaction; brain synchronization; weighted networks
INTRODUCTION
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According to Albert and Barabási [1], the progress that has been made in the study of complex networks has been influenced by at least four factors: the emergence of large databases of the topology of several real networks; an increase in computational power that allows the analysis of networks composed of millions of nodes; the breakdown of the boundaries between different study areas, which has provided researchers with access to numerous databases; finally, the growth of research into general explanations of phenomena, not only from a reductionist perspective but also via an investigation of the relationships between components and the emerging patterns that characterise them. This approach allowed the modeling of several natural phenomena [2,3]. In general, these network models aim to reproduce the dynamic creation of a network and to generate the properties observed in real-world network topologies [4]. However, understanding the mechanisms of a network is not a simple task, since it involves other characteristics such as the process of change of the connection patterns over time: edges may appear and disappear during the lifetime of a network, and there may be heterogeneity between the weights associated with the edges. At the end of the 1990s, Albert and Barabási developed the scale-free network model [5]. This algorithm is composed of the two mechanisms of growth and preferential connection. The first mechanism is related to an increase in the number of nodes of the network at each time step. The second mechanism is defined as a tendency, whereby a new vertex added to the network is connected to a vertex with a high degree. A combination of these mechanisms means that each hub has a high probability of making new connections. A hub is a node whose degree is given by where is the
average degree of the nodes and is the value of the standard deviation. It can therefore be stated that the topologies of these networks involve many nodes with few connections and few very connected hubs, obeying a power law for the degree distribution where . A common feature of the model described above is that the edges of these networks have no weights, and assume a binary structure that indicates only the presence or absence of interaction between the nodes [6]. In unweighted networks, it is not possible to show the diversity and intensity of the connections present in real networks [5, 7]. However, weighted networks offer a natural way to represent this intensity, making it possible to obtain more information about the established connections [8]. Complex networks often exhibit dynamic behaviour in which nodes and edges can be added or removed over time. In order to describe the dynamics of a network, Nicosia et al. [9] developed a mathematical tool called time-varying graphs (TVG) that represents a generalisation of the graph theory formalism. This approach includes the possibility of changes in the properties of the networks, either in the relations between the nodes or in the existence of nodes over the duration of the studied phenomena. Recently, several models have been developed in which weighted networks are considered in terms of their composition. The study in [10] presented a weighted network model for the study of the structure of communities. Using this model, it was possible to study a range of types, from social to biological networks. Barrat et al. [11] proposed a weighted network model in which the structural growth is related to the dynamic evolution of the edge weights. In [12], Li et al. proposed a weighted network model in which the growth of the network is linked to the addition of nodes in the network, and to the connections of these new nodes. Zheng et al. [13] adopted a stochastic scheme for assigning weights to the edges.
Journal Pre-proof
2.1.2) Selecting an edge: We select a vertex, from all vertices connected to to have its connection fortified. The probability, of choosing an edge from the edges connected to vertex , depends on its weight, and the sum of the weights of all the edges connected to vertex , as shown in the following equation: .
(2)
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Once the edge is activated, the intensity of its weight is increased by the parameter . A new vertex is then selected, according to step (i). For each iteration , only one node is selected and one edge is fortified. and are parameters of the model and represent, respectively, the total simulation time and the weight increment for each iteration t. The last step of the model is to subtract 1 from the weight of all edges and rescale the weights between 0 and 1. This eliminate any edges that have not been chosen (zero weight), since the model begins with a full connected network with weight 1 for all edges, as commented before. A TVG structure is used to register the overall model dynamic [9] and the total number of times each edge was activated is recorded in a matrix. A static aggregated network (ASN) [9] is used to represent this matrix. This is a weighted graph in which each weight represents the number of times each edge was fortified. The structure of the TVG was used in the model to facilitate comparison with EEG networks. Figure 1 shows an ASN and its respective TVG.
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In this scheme, both the popularity of a vertex and its aptitude to receive a new vertex are considered. Liu et al. [14] developed a model in which the network growth is linked both to the addition of new nodes and the strengthening of the existing connections in the network. In the model developed by Sun et al. [15], the topological growth of the network is linked to the variation in the weights of its edges. However, none of these authors considered that in some phenomena, the weight of the relationships between nodes is proportional to their occurrence over time, as in the case of the relation between regions in a Brain Functional Network (BFN) [16]. A BFN represents the statistical association patterns between different brain regions. The brain functional connectivity is estimated from physiological signals, such as those obtained from the electroencephalogram (EEG) or functional magnetic resonance imaging (fMRI), by applying methods derived from the Graph Theory and Complex Networks [16,17]. In this paper, we propose a network model that follows similar principles to those of Albert and Barabási’s scale-free networks model [5], called a preferential interaction network (PIN). In a PIN, we have a network of fixed size whose growth is linked to the increase in the weights of its edges and the weight is related to the amount of times it has been chosen in the past. This model produces an algorithm in which edges with larger weights are more likely to receive new connections over time. This characteristic represents the concept of preferential interaction. As a further consideration we compared the PIN model with some EEG based dynamic brain functional networks.
2. METHODOLOGY 2.1 PIN MODEL
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The algorithm that represents the PIN model is divided into two main steps: selection of a node based on its weighted degree, and selection of an edge based on its weight within the node.
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2.1.1) Selecting a node: We start from a fully connected and regular network of degree , where indicates the number of nodes. These initial connections are only used to ensure that all the connections are equally likely to be chosen and will be disregarded at the end of the algorithm. At each instant of time within the total time interval , a vertex is selected based on a probability calculated by the equation (1)
where, is the weight of the edge connecting vertex to vertex . The sum in the numerator is over all the weights of the edges that are connected to the vertex and the sum in the denominator considers all weights of the network, where is a normalized probability. All networks have initial weights that are equal to unity.
FIG. 1: Illustration of how the weights of an ASN network are estimated. A TVG with 5 vertices and ∆t = 3 are represented by ={G(t1), G(t2) and G(t3)}. Each graph G(ti) of the TVG represent a time window in the model. The weight graph GASN is obtained by adding all adjacent matrix of each graph G(ti) within .
In summary, the PIN model is defined by three parameters: the weight addition , the total number of iterations performed in a simulation and the number of nodes in the network . As output, we have a TVG representing the edge activation dynamic and a ASN highlighting the importance over time of each edge. 2.2 BRAIN FUNCTIONAL NETWORKS
In this work, the BFNs were created from the application of the TVG method on the EEG signals of four subjects at resting state, over two minutes. The BFN nodes represent the 30 electrodes of the EEG signals (V = { ; ; ...; }) that were assembled based on the 10-20 system. For each electrode there is an associated time series of the EEG signal,
Journal Pre-proof
This process is performed for a moving time window of 33 ms. By moving the window over the signal period, the process repeatedly generates new networks, thus constructing the time-varying networks. The generated output was a weighted network (ASN) whose edge weights represent the number of times a pair of electrodes remains synchronized over time. The method parameters were as follows: a window size of 33 ms, τ = 5 ms, threshold of 0.9, and λ=1 point [17]. These parameters were chosen considering the results presented in [17].
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The four individuals were between 33 and 66 years old and were healthy; one male and three females. All procedures adopted in the study were approved by the Ethics and Research Committee of the Universidade Federal da Bahia of the Instituto de Ciências da Saúde, under project number CAAE: 444570.1.0000.5662.
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Connectivity between each signal pair was estimated using the Motif-Synchronization (MS) method [17]. The MS method provides a degree of synchronization between the network nodes by counting the quasi-simultaneous occurrence of specific patterns (motifs) within a time series.
The PIN model generates networks with a slower weight probability distribution decay than networks generated from the random addition algorithm, whose slope remains constant, as can be seen in the graphs in Figure 3 (b) and 3 (c). Observing the effects of the weight increment, ∆w, (Figure 3(b)) on the model dynamics, we found that for simulations where ∆w is small compared to the order of the network, ∆w ≪ N, the effect of preferential interaction is smaller and the decay of these net weight distributions is higher. Thus, we obtained networks with a large number of selected edges, preventing the occurrence of edges with larger weights. However, in simulations where a large weight increment is used compared to the order of the network (∆w N) the decay of these distributions decreases and the preferential interaction becomes so dominant that only a few edges are chosen during the simulation, such as is shown by the example in Figure 4(c). The effects of increasing the network size is shown in Figure 3(c). The slope increases as the number of vertices increases. By keeping ∆w fixed and increasing N, the effect of the preferred connection becomes less and less evident. For N >> ∆w, the weight distribution approximates the distribution of a network with randomly selected edges.
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, t ∈ [0; T], where T is the total record time.
3. RESULTS
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For networks of fixed size and constant weight increment, the higher the number of iterations allowed in a simulation, the greater the heterogeneity of the edge weights in the ASN. Figure 2 shows examples for networks with nodes, a weight increment of units and ranging from to iterations.
FIG. 2: Example of ASNs for a simple -node network with a weight increment ∆w of units and ∆t of (a) , (b) and (c) iterations. The thicknesses of the edges are proportional to their weights.
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The graphs in Figure 3 compare the PIN algorithm with a random addition algorithm. In this random algorithm, edges are randomly selected to receive a weight increase of ∆w. Figure 3(a) shows a typical complementary cumulative distribution of weights, P(w), of the model. We simulate each configuration 100 times and estimate the average slope and error on the semi-log P(w) plot, between scales 0.3 and 1.0. For the graph in Figure 3(b) we performed simulations of networks with 100 vertices, iterations, and ∆w ranging between and . Figure 3(c) was constructed analogously, however, we considered networks with N ranging from to , iterations, and ∆w = .
FIG. 3: a) A typical P(w) distribution fitting for ∆w = ,N = and interactions. b) Mean value of the slope modulus of P(w), for networks with 100 vertices, 100 samples, iterations, and ∆w ranging from to c) Average value of the slope modulus of P(w), for networks with the number of vertices ranging from to , 100 samples, iterations, and ∆w = .
Journal Pre-proof number of iterations, since the complementary cumulative probabilities of the weights had high values, as shown in Figure 5(b). Finally, as can be seen in Figures 5(c) and (d), the best fits occurred when we used 800 iterations for subject C and 5000 iterations for D. The increments of the weights were equal to 12 and 3 units for subjects C and D, respectively.
4. APPLICATION
Table I shows the values of the square root of the mean square distance between each subject shown in the graphs in Figure 5 and their respective best fit. According to this table, the fit for Figure 5(a) had the longest distance of 0.035. The distributions in Figure 5(d) showed the smallest value for the distance between the curves of 0.018. The PIN algorithm generates networks in which almost every possible edge has a weight. In comparison, the functional EEG network uses a threshold value to distinguish significant synchronizations from random fluctuations, and therefore generates networks in which only a few edges are selected. The threshold values were obtained using the algorithm proposed by [17] guaranteeing a p-value < 0.05. Thus, for a better comparison of the network topology, in order to keep the both networks at the same size, only the highest weighted edges of the model were considered. Therefore, both PIN and EEG networks have the same number of nodes and edges. In Table I, we compare the network indexes for ASNs from the EEG functional network with the ASNs generated by the best-fit PIN model and a random network with the same number of nodes and edges. The indexes used were the average weighted degree (kw), clustering coefficient (C) and average minimal path length (l).
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We compare the results of the proposed model with those of synchronization electroencephalogram (EEG) networks that evolve over time. The distribution of the edge weights of these networks was compared to the weight distribution of the network edges generated by the model for given values of their parameters. To evaluate which parameters of the model best fitted the physiological edge distributions, we fixed the network order of the model for the same number of the EEG electrodes used. The other parameters, i.e. the number of iterations ( ) and weight increment ( ), were estimated with the aim of minimising the mean quadratic difference between the distributions of the model and subjects. For each set of parameters, 100 simulations were carried out and the mean value was used to fit the model output with the real data. Figure 4 shows the best fit for the complementary cumulative probability of the weights of the edges in the network.
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These results show that as the ratio increases, the variability of the network weights decreases; in this way, the preferential interaction becomes obvious, with only a few edges chosen throughout the simulation. We refer to these highly weighted edges as hub edges.
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FIG. 4: Example SANs of a simple network with nodes and weight increments of (a) 10, (b) and (c) units. The thicknesses of the edges are proportional to their weights.
Based on these results, we can see that depending on the values of the parameters used in the PIN model, it is possible to generate distributions of weights with behaviour that is similar to the distributions of weights of the synchronization networks generated from EEG signals.
FIG. 5: Comparison of the distribution of edge weights between the model and the functional brain networks of healthy subjects at rest. The error bars represent the standard deviation of 100 simulations.
A simulation with 500 iterations and a weight increment of 11 units was the best fit for subject A, as can be seen in Figure 5(a). For subject B, it was necessary to use a larger
TABLE I: Weighted average degree , clustering coefficient ( ), minimal path length (l) for the EEG network, PIN model and a random model. Also is presented the RMS between the EEG network and the PIN model.
Journal Pre-proof [6] Y. Lin, H. Xiong, M. Chen, X. Sun, E. Feng, J. Liu, and B. Wang, Kybernetes 41, 1244 (2012). [7] M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, Physica A: Statistical Mechanics and its Applications 346, 34 (2005). [8] M. Li, J. Wu, D. Wang, T. Zhou, Z. Di, and Y. Fan, Physica A: Statistical Mechanics and its Applications 375, 355 (2007). [9] V. Nicosia, J. Tang, M. Musolesi, G. Russo, C. Mascolo, and V. Latora, Chaos: An Interdiciplinary Journal of Nonlinear Science 22, 023101 (2012).
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Despite the simplicity of the model and the complexity of the functional brain networks, the indices shown in Table 1 shows that the PIN model is more comparable with the EEG network than any random network generated using the same number of nodes, edges and iterations, particularly for subject D, in which the RMS value is lower. However, despite the accuracy of the P(w) curve fitting, the indices of the model have systematically lower values than the indices of the EEG networks. This may indicate that the preferential interaction attachment mechanism is not enough in forming the final network topology. In this case, the PIN model would better describe the dynamic mechanism of choosing edges, lacking spatial dependence mechanisms that could increase the clustering and the minimum path length of the model.
[10] C. Li and G. Chen, Physica A: Statistical Mechanics and its Applications 370, 869 (2006).
5. DISCUSSION
[12] P. Li, Q. Zhao, and H. Wang, International Journal of Modern Physics B 27, 1350039 (2013). [13] D. Zheng, S. Trimper, B. Zheng, and P. Hui, Physical Review E 67, 040102 (2003). [14] J.-G. Liu, Y.-Z. Dang, W.-X. Wang, Z.-T. Wang, T. Zhou, B.-H. Wang, Q. Guo, Z.-G. Xuan, S.-H. Jiang, and M.-W. Zhao, arXiv preprint physics/0512270 (2005).
Acknowledgments
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Evaluating the interpretation of time in the model, we can relate the number of iterations ( ) in the best-fit model as an individual characteristic of the brain dynamics. This characteristic could be related to the formation velocity of the synchronization patterns. The networks constructed from EEG signals represent the connectivity patterns of the brain synchronization of individuals, and since these patterns are distinct for each individual, they can be said to have intrinsic individual variability. Therefore, networks constructed from the EEG signals show a different topology for each individual. It is believed that the unique patterns of cerebral connectivity are related to the connections present in the brain of a single individual. These connections may have existed since its formation. By using the PIN model to describe these connectivity patterns in the EEG data, we show that it is possible to reproduce how the brain connections were made, and to form the connectivity pattern of a particular subject. Thus, we suggest that the patterns of connectivity of brain synchronization, as measured via an EEG, can be reproduced when we consider a process based on preferential interaction dynamics.
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[11] A. Barrat, M. Barthélemy, and A. Vespignani, Physical Review E 70, 066149 (2004).
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES)- Finance Code 001.
References
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[1] M. E. J. Newman, Networks, vol. volume nico (OUP Oxford, New York, 2010), 1st ed. [2] S. N. Dorogovtsev and J. F. Mendes, Evolution of networks: From biological nets to the Internet and WWW (OUP Oxford, 2013). [3] R. Albert and A.-L. Barabási, Reviews of Modern Physics 74, 47 (2002). [4] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Physics Reports 424, 175 (2006). [5] A.-L. Barabási and R. Albert, 286, 509 (1999).
[15] X. Sun, E. Feng, J. Liu, and B. Wang, Kybernetes 41, 1244 (2012). [16] P. T. Cardoso Canário, Mestrado em física, Universidade Federal da Bahia, Salvador (2013). [17] R. Rosário, P. Cardoso, M. Munõz, P. Montoya, and J. Miranda, Physica A: Statistical Mechanics and its Applications 439, 7 (2015).
Journal Pre-proof Table
D
0.61
2.37
PIN
0.89 ± 0.23
0.21 ± 0.03
2.01 ± 0.07
Rand
6.17 ± 0.83
0.71 ± 0.01
1.29 ± 0.01
EEG
2.09
0.6
2.26
PIN
0.95 ± 0.24
0.27 ± 0.02
1.8 ± 0.5
Rand
8.47 ± 0.94
0.92 ± 0.01
1.08 ± 0.01
EEG
1.29
0.54
2.28
PIN
0.85 ± 0.22
0.23 ± 0.03
Rand
7.8 ± 0.87
0.86 ± 0.01
EEG
2.8
0.81
PIN
2.1 ± 0.4
0.70 ± 0.02
Rand
14.4 ± 1.0
1.0 ± 0
1
0.035
1.92 ± 0.06
0.022
0.029
1.14 ± 0.01 1.3
1.29 ± 0.02 1.0 ± 0
of
2.58
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EEG
RMS
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C
l
al
B
C
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A
kw
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Subject Index
0.018
Journal Pre-proof *Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
PII: DOI: Reference:
S0378-4371(20)30070-4 https://doi.org/10.1016/j.physa.2020.124259 PHYSA 124259
To appear in:
Physica A
Received date : 1 April 2019 Revised date : 30 December 2019 Please cite this article as: R.A. Sousa, V.N.A. Lula-Rocha, T. Toutain et al., Preferential interaction networks: A dynamic model for brain synchronization networks, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2020.124259. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
*Highlights (for review)
Journal Pre-proof Highlights:
of
p ro
Pr e-
al
urn
A network computational model testing the hypothesis of preferential interaction was created. The dependence of the model on the weight increment and network size and their limits were evaluated. The electroencephalogram (EEG) of four healthy subjects were collected to be used as model validation. The motif synchronization method was applied to the EEG data generating weighted synchronization networks. The network weight distribution, generated by the model, were compared with those generated by the EEG.
Jo
*Manuscript Click here to view linked References
Journal Pre-proof Preferential interaction networks: A dynamic model for brain synchronization networks 1
1
2
1
2
R.A. Sousa, V.N.A Lula-Rocha, T. Toutain, R.S. Rosário, E.C.B. Cambui, and J.G.V. Miranda 1 2
1,
Institute of Physics, Federal University of Bahia (IF-UFBA), 40210-340 Salvador, Brazil
Institute of Biology, Federal University of Bahia (IBIO-UFBA), 40,170-110 Salvador, Brazil
p ro
of
In this work, we propose a weighted network model called a preferential interaction network (PIN), whose construction is based on mechanisms similar to those of the scalefree network model developed by Albert and Barabási. The PIN aims to reproduce the behaviour of real systems of fixed size in which stronger connections are reinforced over time. Its composition starts with a network with fixed nodes, and obeys two basic processes: firstly, an increase in the edge weights, and secondly, a preferential interaction is added. Preferential interaction is defined as the tendency whereby edges with larger weights are more likely to be chosen in a iteration. PIN has three parameters: the size of the network, the rate of increase of the weights, and the number of iterations. These parameters were used to fit the model to EEG brain synchronization networks of healthy subjects. The complementary cumulative distribution of weight in the model was compared to the corresponded distribution in EEG networks. The shape of the model distribution showed similar to those of the EEG data networks and the PIN model topology index are closer than a random null model, suggesting that a preferential interaction process represents the core of brain synchronization dynamics.
Pr e-
Keywords: complex networks; preferential interaction; brain synchronization; weighted networks
INTRODUCTION
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al
According to Albert and Barabási [1], the progress that has been made in the study of complex networks has been influenced by at least four factors: the emergence of large databases of the topology of several real networks; an increase in computational power that allows the analysis of networks composed of millions of nodes; the breakdown of the boundaries between different study areas, which has provided researchers with access to numerous databases; finally, the growth of research into general explanations of phenomena, not only from a reductionist perspective but also via an investigation of the relationships between components and the emerging patterns that characterise them. This approach allowed the modeling of several natural phenomena [2,3]. In general, these network models aim to reproduce the dynamic creation of a network and to generate the properties observed in real-world network topologies [4]. However, understanding the mechanisms of a network is not a simple task, since it involves other characteristics such as the process of change of the connection patterns over time: edges may appear and disappear during the lifetime of a network, and there may be heterogeneity between the weights associated with the edges. At the end of the 1990s, Albert and Barabási developed the scale-free network model [5]. This algorithm is composed of the two mechanisms of growth and preferential connection. The first mechanism is related to an increase in the number of nodes of the network at each time step. The second mechanism is defined as a tendency, whereby a new vertex added to the network is connected to a vertex with a high degree. A combination of these mechanisms means that each hub has a high probability of making new connections. A hub is a node whose degree is given by where is the
average degree of the nodes and is the value of the standard deviation. It can therefore be stated that the topologies of these networks involve many nodes with few connections and few very connected hubs, obeying a power law for the degree distribution where . A common feature of the model described above is that the edges of these networks have no weights, and assume a binary structure that indicates only the presence or absence of interaction between the nodes [6]. In unweighted networks, it is not possible to show the diversity and intensity of the connections present in real networks [5, 7]. However, weighted networks offer a natural way to represent this intensity, making it possible to obtain more information about the established connections [8]. Complex networks often exhibit dynamic behaviour in which nodes and edges can be added or removed over time. In order to describe the dynamics of a network, Nicosia et al. [9] developed a mathematical tool called time-varying graphs (TVG) that represents a generalisation of the graph theory formalism. This approach includes the possibility of changes in the properties of the networks, either in the relations between the nodes or in the existence of nodes over the duration of the studied phenomena. Recently, several models have been developed in which weighted networks are considered in terms of their composition. The study in [10] presented a weighted network model for the study of the structure of communities. Using this model, it was possible to study a range of types, from social to biological networks. Barrat et al. [11] proposed a weighted network model in which the structural growth is related to the dynamic evolution of the edge weights. In [12], Li et al. proposed a weighted network model in which the growth of the network is linked to the addition of nodes in the network, and to the connections of these new nodes. Zheng et al. [13] adopted a stochastic scheme for assigning weights to the edges.
Journal Pre-proof
2.1.2) Selecting an edge: We select a vertex, from all vertices connected to to have its connection fortified. The probability, of choosing an edge from the edges connected to vertex , depends on its weight, and the sum of the weights of all the edges connected to vertex , as shown in the following equation: .
(2)
p ro
of
Once the edge is activated, the intensity of its weight is increased by the parameter . A new vertex is then selected, according to step (i). For each iteration , only one node is selected and one edge is fortified. and are parameters of the model and represent, respectively, the total simulation time and the weight increment for each iteration t. The last step of the model is to subtract 1 from the weight of all edges and rescale the weights between 0 and 1. This eliminate any edges that have not been chosen (zero weight), since the model begins with a full connected network with weight 1 for all edges, as commented before. A TVG structure is used to register the overall model dynamic [9] and the total number of times each edge was activated is recorded in a matrix. A static aggregated network (ASN) [9] is used to represent this matrix. This is a weighted graph in which each weight represents the number of times each edge was fortified. The structure of the TVG was used in the model to facilitate comparison with EEG networks. Figure 1 shows an ASN and its respective TVG.
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In this scheme, both the popularity of a vertex and its aptitude to receive a new vertex are considered. Liu et al. [14] developed a model in which the network growth is linked both to the addition of new nodes and the strengthening of the existing connections in the network. In the model developed by Sun et al. [15], the topological growth of the network is linked to the variation in the weights of its edges. However, none of these authors considered that in some phenomena, the weight of the relationships between nodes is proportional to their occurrence over time, as in the case of the relation between regions in a Brain Functional Network (BFN) [16]. A BFN represents the statistical association patterns between different brain regions. The brain functional connectivity is estimated from physiological signals, such as those obtained from the electroencephalogram (EEG) or functional magnetic resonance imaging (fMRI), by applying methods derived from the Graph Theory and Complex Networks [16,17]. In this paper, we propose a network model that follows similar principles to those of Albert and Barabási’s scale-free networks model [5], called a preferential interaction network (PIN). In a PIN, we have a network of fixed size whose growth is linked to the increase in the weights of its edges and the weight is related to the amount of times it has been chosen in the past. This model produces an algorithm in which edges with larger weights are more likely to receive new connections over time. This characteristic represents the concept of preferential interaction. As a further consideration we compared the PIN model with some EEG based dynamic brain functional networks.
2. METHODOLOGY 2.1 PIN MODEL
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The algorithm that represents the PIN model is divided into two main steps: selection of a node based on its weighted degree, and selection of an edge based on its weight within the node.
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2.1.1) Selecting a node: We start from a fully connected and regular network of degree , where indicates the number of nodes. These initial connections are only used to ensure that all the connections are equally likely to be chosen and will be disregarded at the end of the algorithm. At each instant of time within the total time interval , a vertex is selected based on a probability calculated by the equation (1)
where, is the weight of the edge connecting vertex to vertex . The sum in the numerator is over all the weights of the edges that are connected to the vertex and the sum in the denominator considers all weights of the network, where is a normalized probability. All networks have initial weights that are equal to unity.
FIG. 1: Illustration of how the weights of an ASN network are estimated. A TVG with 5 vertices and ∆t = 3 are represented by ={G(t1), G(t2) and G(t3)}. Each graph G(ti) of the TVG represent a time window in the model. The weight graph GASN is obtained by adding all adjacent matrix of each graph G(ti) within .
In summary, the PIN model is defined by three parameters: the weight addition , the total number of iterations performed in a simulation and the number of nodes in the network . As output, we have a TVG representing the edge activation dynamic and a ASN highlighting the importance over time of each edge. 2.2 BRAIN FUNCTIONAL NETWORKS
In this work, the BFNs were created from the application of the TVG method on the EEG signals of four subjects at resting state, over two minutes. The BFN nodes represent the 30 electrodes of the EEG signals (V = { ; ; ...; }) that were assembled based on the 10-20 system. For each electrode there is an associated time series of the EEG signal,
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This process is performed for a moving time window of 33 ms. By moving the window over the signal period, the process repeatedly generates new networks, thus constructing the time-varying networks. The generated output was a weighted network (ASN) whose edge weights represent the number of times a pair of electrodes remains synchronized over time. The method parameters were as follows: a window size of 33 ms, τ = 5 ms, threshold of 0.9, and λ=1 point [17]. These parameters were chosen considering the results presented in [17].
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The four individuals were between 33 and 66 years old and were healthy; one male and three females. All procedures adopted in the study were approved by the Ethics and Research Committee of the Universidade Federal da Bahia of the Instituto de Ciências da Saúde, under project number CAAE: 444570.1.0000.5662.
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Connectivity between each signal pair was estimated using the Motif-Synchronization (MS) method [17]. The MS method provides a degree of synchronization between the network nodes by counting the quasi-simultaneous occurrence of specific patterns (motifs) within a time series.
The PIN model generates networks with a slower weight probability distribution decay than networks generated from the random addition algorithm, whose slope remains constant, as can be seen in the graphs in Figure 3 (b) and 3 (c). Observing the effects of the weight increment, ∆w, (Figure 3(b)) on the model dynamics, we found that for simulations where ∆w is small compared to the order of the network, ∆w ≪ N, the effect of preferential interaction is smaller and the decay of these net weight distributions is higher. Thus, we obtained networks with a large number of selected edges, preventing the occurrence of edges with larger weights. However, in simulations where a large weight increment is used compared to the order of the network (∆w N) the decay of these distributions decreases and the preferential interaction becomes so dominant that only a few edges are chosen during the simulation, such as is shown by the example in Figure 4(c). The effects of increasing the network size is shown in Figure 3(c). The slope increases as the number of vertices increases. By keeping ∆w fixed and increasing N, the effect of the preferred connection becomes less and less evident. For N >> ∆w, the weight distribution approximates the distribution of a network with randomly selected edges.
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, t ∈ [0; T], where T is the total record time.
3. RESULTS
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For networks of fixed size and constant weight increment, the higher the number of iterations allowed in a simulation, the greater the heterogeneity of the edge weights in the ASN. Figure 2 shows examples for networks with nodes, a weight increment of units and ranging from to iterations.
FIG. 2: Example of ASNs for a simple -node network with a weight increment ∆w of units and ∆t of (a) , (b) and (c) iterations. The thicknesses of the edges are proportional to their weights.
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The graphs in Figure 3 compare the PIN algorithm with a random addition algorithm. In this random algorithm, edges are randomly selected to receive a weight increase of ∆w. Figure 3(a) shows a typical complementary cumulative distribution of weights, P(w), of the model. We simulate each configuration 100 times and estimate the average slope and error on the semi-log P(w) plot, between scales 0.3 and 1.0. For the graph in Figure 3(b) we performed simulations of networks with 100 vertices, iterations, and ∆w ranging between and . Figure 3(c) was constructed analogously, however, we considered networks with N ranging from to , iterations, and ∆w = .
FIG. 3: a) A typical P(w) distribution fitting for ∆w = ,N = and interactions. b) Mean value of the slope modulus of P(w), for networks with 100 vertices, 100 samples, iterations, and ∆w ranging from to c) Average value of the slope modulus of P(w), for networks with the number of vertices ranging from to , 100 samples, iterations, and ∆w = .
Journal Pre-proof number of iterations, since the complementary cumulative probabilities of the weights had high values, as shown in Figure 5(b). Finally, as can be seen in Figures 5(c) and (d), the best fits occurred when we used 800 iterations for subject C and 5000 iterations for D. The increments of the weights were equal to 12 and 3 units for subjects C and D, respectively.
4. APPLICATION
Table I shows the values of the square root of the mean square distance between each subject shown in the graphs in Figure 5 and their respective best fit. According to this table, the fit for Figure 5(a) had the longest distance of 0.035. The distributions in Figure 5(d) showed the smallest value for the distance between the curves of 0.018. The PIN algorithm generates networks in which almost every possible edge has a weight. In comparison, the functional EEG network uses a threshold value to distinguish significant synchronizations from random fluctuations, and therefore generates networks in which only a few edges are selected. The threshold values were obtained using the algorithm proposed by [17] guaranteeing a p-value < 0.05. Thus, for a better comparison of the network topology, in order to keep the both networks at the same size, only the highest weighted edges of the model were considered. Therefore, both PIN and EEG networks have the same number of nodes and edges. In Table I, we compare the network indexes for ASNs from the EEG functional network with the ASNs generated by the best-fit PIN model and a random network with the same number of nodes and edges. The indexes used were the average weighted degree (kw), clustering coefficient (C) and average minimal path length (l).
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We compare the results of the proposed model with those of synchronization electroencephalogram (EEG) networks that evolve over time. The distribution of the edge weights of these networks was compared to the weight distribution of the network edges generated by the model for given values of their parameters. To evaluate which parameters of the model best fitted the physiological edge distributions, we fixed the network order of the model for the same number of the EEG electrodes used. The other parameters, i.e. the number of iterations ( ) and weight increment ( ), were estimated with the aim of minimising the mean quadratic difference between the distributions of the model and subjects. For each set of parameters, 100 simulations were carried out and the mean value was used to fit the model output with the real data. Figure 4 shows the best fit for the complementary cumulative probability of the weights of the edges in the network.
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These results show that as the ratio increases, the variability of the network weights decreases; in this way, the preferential interaction becomes obvious, with only a few edges chosen throughout the simulation. We refer to these highly weighted edges as hub edges.
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FIG. 4: Example SANs of a simple network with nodes and weight increments of (a) 10, (b) and (c) units. The thicknesses of the edges are proportional to their weights.
Based on these results, we can see that depending on the values of the parameters used in the PIN model, it is possible to generate distributions of weights with behaviour that is similar to the distributions of weights of the synchronization networks generated from EEG signals.
FIG. 5: Comparison of the distribution of edge weights between the model and the functional brain networks of healthy subjects at rest. The error bars represent the standard deviation of 100 simulations.
A simulation with 500 iterations and a weight increment of 11 units was the best fit for subject A, as can be seen in Figure 5(a). For subject B, it was necessary to use a larger
TABLE I: Weighted average degree , clustering coefficient ( ), minimal path length (l) for the EEG network, PIN model and a random model. Also is presented the RMS between the EEG network and the PIN model.
Journal Pre-proof [6] Y. Lin, H. Xiong, M. Chen, X. Sun, E. Feng, J. Liu, and B. Wang, Kybernetes 41, 1244 (2012). [7] M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, Physica A: Statistical Mechanics and its Applications 346, 34 (2005). [8] M. Li, J. Wu, D. Wang, T. Zhou, Z. Di, and Y. Fan, Physica A: Statistical Mechanics and its Applications 375, 355 (2007). [9] V. Nicosia, J. Tang, M. Musolesi, G. Russo, C. Mascolo, and V. Latora, Chaos: An Interdiciplinary Journal of Nonlinear Science 22, 023101 (2012).
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Despite the simplicity of the model and the complexity of the functional brain networks, the indices shown in Table 1 shows that the PIN model is more comparable with the EEG network than any random network generated using the same number of nodes, edges and iterations, particularly for subject D, in which the RMS value is lower. However, despite the accuracy of the P(w) curve fitting, the indices of the model have systematically lower values than the indices of the EEG networks. This may indicate that the preferential interaction attachment mechanism is not enough in forming the final network topology. In this case, the PIN model would better describe the dynamic mechanism of choosing edges, lacking spatial dependence mechanisms that could increase the clustering and the minimum path length of the model.
[10] C. Li and G. Chen, Physica A: Statistical Mechanics and its Applications 370, 869 (2006).
5. DISCUSSION
[12] P. Li, Q. Zhao, and H. Wang, International Journal of Modern Physics B 27, 1350039 (2013). [13] D. Zheng, S. Trimper, B. Zheng, and P. Hui, Physical Review E 67, 040102 (2003). [14] J.-G. Liu, Y.-Z. Dang, W.-X. Wang, Z.-T. Wang, T. Zhou, B.-H. Wang, Q. Guo, Z.-G. Xuan, S.-H. Jiang, and M.-W. Zhao, arXiv preprint physics/0512270 (2005).
Acknowledgments
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Evaluating the interpretation of time in the model, we can relate the number of iterations ( ) in the best-fit model as an individual characteristic of the brain dynamics. This characteristic could be related to the formation velocity of the synchronization patterns. The networks constructed from EEG signals represent the connectivity patterns of the brain synchronization of individuals, and since these patterns are distinct for each individual, they can be said to have intrinsic individual variability. Therefore, networks constructed from the EEG signals show a different topology for each individual. It is believed that the unique patterns of cerebral connectivity are related to the connections present in the brain of a single individual. These connections may have existed since its formation. By using the PIN model to describe these connectivity patterns in the EEG data, we show that it is possible to reproduce how the brain connections were made, and to form the connectivity pattern of a particular subject. Thus, we suggest that the patterns of connectivity of brain synchronization, as measured via an EEG, can be reproduced when we consider a process based on preferential interaction dynamics.
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[11] A. Barrat, M. Barthélemy, and A. Vespignani, Physical Review E 70, 066149 (2004).
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES)- Finance Code 001.
References
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[1] M. E. J. Newman, Networks, vol. volume nico (OUP Oxford, New York, 2010), 1st ed. [2] S. N. Dorogovtsev and J. F. Mendes, Evolution of networks: From biological nets to the Internet and WWW (OUP Oxford, 2013). [3] R. Albert and A.-L. Barabási, Reviews of Modern Physics 74, 47 (2002). [4] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Physics Reports 424, 175 (2006). [5] A.-L. Barabási and R. Albert, 286, 509 (1999).
[15] X. Sun, E. Feng, J. Liu, and B. Wang, Kybernetes 41, 1244 (2012). [16] P. T. Cardoso Canário, Mestrado em física, Universidade Federal da Bahia, Salvador (2013). [17] R. Rosário, P. Cardoso, M. Munõz, P. Montoya, and J. Miranda, Physica A: Statistical Mechanics and its Applications 439, 7 (2015).
Journal Pre-proof Table
D
0.61
2.37
PIN
0.89 ± 0.23
0.21 ± 0.03
2.01 ± 0.07
Rand
6.17 ± 0.83
0.71 ± 0.01
1.29 ± 0.01
EEG
2.09
0.6
2.26
PIN
0.95 ± 0.24
0.27 ± 0.02
1.8 ± 0.5
Rand
8.47 ± 0.94
0.92 ± 0.01
1.08 ± 0.01
EEG
1.29
0.54
2.28
PIN
0.85 ± 0.22
0.23 ± 0.03
Rand
7.8 ± 0.87
0.86 ± 0.01
EEG
2.8
0.81
PIN
2.1 ± 0.4
0.70 ± 0.02
Rand
14.4 ± 1.0
1.0 ± 0
1
0.035
1.92 ± 0.06
0.022
0.029
1.14 ± 0.01 1.3
1.29 ± 0.02 1.0 ± 0
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2.58
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EEG
RMS
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C
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B
C
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Subject Index
0.018
Journal Pre-proof *Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: