Azimuthal Anisotropy in High Energy Nuclear Collision - An Approach based on Complex Network Analysis

Recently, a complex network based method of Visibility Graph has been applied to confirm the scale-freeness and presence of fractal properties in the process of multiplicity fluctuation. Analysis of data obtained from experiments on hadron-nucleus and nucleus-nucleus interactions results in values of Power-of-Scale-freeness-of-Visibility-Graph-(PSVG) parameter extracted from the visibility graphs. Here, the relativistic nucleus-nucleus interaction data have been analysed to detect azimuthal-anisotropy by extending the Visibility Graph method and extracting the average clustering coefficient, one of the important topological parameters, from the graph. Azimuthal-distributions corresponding to different pseudorapidity-regions around the central-pseudorapidity value are analysed utilising the parameter. Here we attempt to correlate the conventional physical significance of this coefficient with respect to complex-network systems, with some basic notions of particle production phenomenology, like clustering and correlation. Earlier methods for detecting anisotropy in azimuthal distribution, were mostly based on the analysis of statistical fluctuation. In this work, we have attempted to find deterministic information on the anisotropy in azimuthal distribution by means of precise determination of topological parameter from a complex network perspective.


Introduction
Number of authors have probed the azimuthal anisotropy of the produced particles in ultra-relativistic heavy-ion collisions as a function of transverse momentum as one of the observables to study the collective properties of nuclear matter [Ex. [3]]. The primary volume holding the interacting nuclear substance is essentially anisotropic in coordinate space, because of the geometry present in noncentral heavy-ion collisions. The framework in which the produced nuclear matter thermalize in the anisotropic volume, resulting its initial anisotropy from the coordinate space, to be transmitted via reciprocal interactions into the resultant observable anisotropy in momentum space, had been a field of immense interest. In various literatures, this phenomenon has been referred as collective anisotropic flow which has been considered as direct examination of the degree of thermalization of the produced matter, and an indirect one of its properties like viscosity etc.
To analyse the underlying cause of the collective anisotropic flow, the azimuthal anisotropic distribution in momentum space has been analysed with Fourier series [4], where first few harmonicas has been referred to as direct flow, elliptical flow etc. in the symmetry plane. However from these harmonics, it could not be confirmed that the azimuthal anisotropic distribution in momentum space has been resulted from collective anisotropic flow or from some fully unrelated physical process yielding event-by-event anisotropies, like mini-jets. Then, to analyse collectivity more rigorously, use of correlation techniques by including two or more particles has evolved and eventually multi-particle correlation techniques have evolved. Recently, Bilandzic et. al. have suggested that if all produced particles are independently discharged, then the presence of collective anisotropic flow can be confirmed [5]. This had been confirmed mathematically in [6]. Sarkisyan [7] has analysed multiplicity distribution of the produced hadrons in high-energy interaction to delineate high-order genuine correlations [8] and established the perquisite of incorporating the multiparticle property of the correlations with the property of self-similarity to achieve a good description of the measurements. For more than three decades, this multi-particle correlation methods have been applied to analyse anisotropic flow. Wang et.al. [9] and Jiang et. al. [10] were the first to go beyond two-particle azimuthal correlations, in terms of experimental analysis. However, it did not work for increased number of particles in such multiplets. The joint probability distribution of M number of particles with an M -multiplicity event has been applied theoretically, for the first time, in flow analysis of global event shapes [6] and then in other further studies [Ex. [3]]. Borghini et. al. have further reported a series of analysis on multiparticle correlations and cumulants [11]. Two and multiparticle cumulants had drawbacks stemming from trivial and non-negligible contributions from autocorrelations, which generated interference among the various harmonics. Then Lee-Yang zero(LYZ) method [12,13] was introduced, which separates the authentic multi-particle estimate for flow harmonics, equivalent to the asymptotic behaviour of the cumulant series but this approach has its own integral systematic biases. Most recently, Bilandzic et. al. have proposed Q-cumulants by implementing Voloshin's fundamental idea of manifesting multi-particle azimuthal correlations in terms of Q-vectors assessed for different harmonics [14]. Previous drawbacks were partly removed in this method but is very monotonous to calculate and hence could be accomplished only for a small subset of multi-particle azimuthal correlations. Then, Bilandzic et. al. have provided a generic framework which allows all multi-particle azimuthal correlations to be evaluated analytically, with a fast single pass over the data [5]. It removed previous limitations and new multi-particle azimuthal observables could be materialised experimentally but in this method, systematic bias has been found in conventional differential flow analyses, when all particles got divided into the two groups -one of reference particles and another particles of interest.
Study of flow analysis can also be attempted using different methods based on fractality of a complex system, as the analysis of fractal characteristics of fluctuations in high energy interactions has become a potential approach since the last decade[Eg. [15,16,17,18,19,20,21]].
Recently, novel approaches have been proposed to interpret complex systems by means of complex networks. Various natural systems, can be termed as complex, heterogeneous systems consisting of various kinds of fundamental units which communicate among themselves through varied interactions(viz. long-range and short range). Complex network-based systems present us a quantitative model for large-scale natural systems(in the various fields of physics, biology and the social sciences). The topological parameters extracted from these complex networks derived from the real systems give important information about the nature of the real system. The latest advances in the field of complex network, have been reviewed and the analytical models for random graphs, small-world and scale-free networks have been analysed by Albert and Barabá, in the recent past [22,23]. Havlin et al. have reported the relevance of network sciences to the analysis, perception, design and repair of multi-level complex systems which are discovered in man-made and human social systems, in organic and inorganic matter, in various scales(from nano to macro), and in natural and anthropogenic structures [24]. Using the heterogeneous time scales, Zhao et al. have examined the dynamics of stock market, using correlation-based network and detected global expansion and local clustering market behaviours during crises [25].
Lacasa et al. have introduced a very interesting method of Visibility Graph analysis [26,27] which has gained importance because its completely different, rigorous approach to estimate fractality. They have started applying the classical method of complex network analysis to measure long-range dependence and fractality of a time series [27]. Using fractional Brownian motion(fBm) and fractional Gaussian noises(fGn) series as a theoretical framework, Lacasa et al. have experimented over real time series in various scientific fields. Lacasa et al. [27] converted fractional Brownian motion(fBm) and fractional Gaussian noises(fGn) series into a scale-free Visibility Graph having degree distribution as a function of the Hurst parameter(associated with Fractal Dimension which is the degree of fractality of the time-series and deduced from the Detrended Fluctuation Analysis(DFA) function of the time-series [28]). Recently, multiplicity fluctuation in (π − -AgBr(350 GeV)) and ( 32 S-AgBr(200 A GeV)) interactions has been analysed using Visibility Graph method [1,2] and the fractality of void probability distribution in 32 S-Ag/Br interaction at an incident energy of 200 GeV per nucleon has also been recently, using this method [29,30,31,32].
Motivated by the inspiring findings acquired from the

Visibility Graph Method
previous studies using Visibility Graph approach, in this work, the azimuthal distribution data for an exemplary interaction -32 S-AgBr(200 A GeV) has been analysed extensively for the presence of collective anisotropic flow by extending the complex network based Visibility Graph approach. The Average-clustering-co-efficient [33] -one of the important topological parameters, is extracted from the Visibility Graph constructed from the azimuthal distribution data corresponding to each of the pseudorapidity regions around the central rapidity and then, analysed further. The scale-freeness and the fractal properties of the process of pionisation, have already been confirmed in [1,2,29,30,31,32] by experimenting on the specimen data of (π − -AgBr(350 GeV)) and ( 32 S-AgBr(200 A GeV)) interactions and then, by analysing the values Power of Scale-freeness of Visibility Graph -PSVG [26,27,34] parameter extracted from the Graphs constructed from the data. The topological parameters of the Visibility graphs have their the usual significance with respect to complex network systems. Here we attempted to correlate the physical significance of Average-clustering-co-efficient with some fundamental notions of particle production phenomenology, like clustering and correlation. Earlier methods for detecting anisotropy in azimuthal distribution, were mostly based on the analysis of statistical fluctuation. So, in this work, we have attempted to analyse collective anisotropic flow by analysing the azimuthal distribution using a poles apart, rigorous approach of complex network perspective which gives more deterministic information about the anisotropy in azimuthal distribution by means of precise topological parameters. The rest of the paper is organized as follows. The method of Visibility Graph algorithm and the significance of complex network parameters like scale-freeness property, Average-clustering-co-efficient are presented in Section 2. The data description and related terminologies are elaborated in Section 3.1. The details of our analysis and the inferences from the test results are given in Section 3.2. The physical significance of the network parameter and its prospective correlation with the traditional concepts of collective anisotropic flow in heavy-ion collisions, is elaborated and the paper is concluded in Section 4.

Method of analysis
As per the Visibility Graph method, a graph is formed a time or data series according to the visibility of each node from the rest of the nodes [27]. This way the Visibility Graph preserves the dynamics of the fluctuation of the data series inside it. Hence periodic series is transformed to regular graph, random series to random graph and naturally fractal series to scale-free network in which the graph's degree distribution conforms to the power-law with respect to its degree. This way a fractal series can be mapped into scale-free Visibility Graph [27], that too from series with finite number of points [35] as against the other non-stationary, nonlinear methods like DFA, MF-DFA which mandates infinite data as input.

Visibility Graph Method
Let's suppose that the value of i t h(in the sequence of the input data series) point of time series is X i . This way all the input data points are mapped to their corresponding nodes or vertices(according to their value or magnitude). In this node-series, two nodes, say X m and X n corresponding to m t h and n t h points in the time series, are said to be visible to each other or in other words joined by a two-way edge, if and only if, the equation 1 is satisfied. This way, the Visibility Graph is constructed out of a time series X.
where ∀j ∈ Z + and j < (n − m) The nodes X m and X n with m = i and n = i+6 are shown in the Fig. 1 where the nodes X m and X n are visible to each other and connected with a bi-directional edge as the Eq. 1 is valid for them. It is evident that sequential nodes are always connected as two sequential points of the time or data series can always see each other. The degree of a node of the graph -here Visibility Graph, is the number of connections/edges the node has with the rest of the nodes in the graph. Hence, the degree distribution of a network as P (k) is denoted as the fraction of nodes in the network, having degree k. Let's assume that there are n k number of nodes with degree k and n is the total number of nodes in a network, then P (k) = n k /n for all probable k-s.
As per Lacasa et al. [26,27] and Ahmadlou et al. [34], the degree of scale-freeness of Visibility Graph corresponds to the degree of fractality and complexity of the input time or data series. The manifestation of the scale-freeness property of a Visibility Graph, is reflected in its degree distribution, if it follows the power-law. It means, if for the Visibility Graph, P (k) ∼ k −λp , where λ p , a constant, is then called the Power of the Scale-freeness in Visibility Graph or PSVG. PSVG, thus signifies the degree of selfsimilarity, fractality and a measurement of complexity of  [26,27,34]. Also, there is an inverse linear relationship between PSVG and Hurst exponent measured time series [27].

Average-clustering-co-efficient of Graph(Here Visibility Graph)
As we know that by definition, a cluster in a network is a set of nodes with similar features. In this experiment, we have extracted clusters based on Density-Based Algorithm proposed by Ester et al. [36] from the Visibility networks constructed from various datasets. Here, the density of nodes has been measured between each pair of visible nodes in the Visibility Graph in terms of number of nodes(which has a threshold value -let's denote by δ in our experiment), to create clusters. For each node(let's denote by n a ), among all the nodes visible from n a , the ones(let's denote by [n b1 , n b2 , . . . , n bn ]) having the count of nodes above δ, between [n b1 , n b2 , . . . , n bn ] and the source node-n a , remain in the same cluster. For each of the datasets corresponding to each cluster, Visibility Graphs are constructed and the Average-clustering-coefficient is calculated.
Clustering coefficient is the calculation of the extent to which nodes in a graph has the tendency to cluster together. Average-clustering-co-efficient has been defined by Watts and Strogatz [33], as the overall clustering coefficient of a network, which is estimated as the mean local clustering coefficient of all the nodes in the network. High value of this co-efficient is indicates the robustness of a network.

Experimental details 3.1 Data description
The experimental data has been acquired by exposing Illford G5 emulsion plates to a 32 S-beam of 200 GeV incident energy per nucleon, from CERN. A Leitz Metaloplan microscope having 10X ocular lens and equipped with an additional semi-automatic scanning stage, was used to examine the plate. Each plate was examined by two individualistic observers to increase the accuracy in detection, counting and measurement procedure. This way, the scanning efficiency could be maximised. For the final quantification, an oil immersion-10X objective was used.
The measuring system has specification of 1µm resolution along X and Y axes, and 0.5µm resolution along Z axis.
In our previous works [37,38,39,40,41,42,43] the basis for event selection is already explained. In the context of nuclear emulsion [44], after interactions, the discharged particle are categorized as shower, grey and black particles.

Our method of analysis
In the work [1], the pseudorapidity(η)-space for 10 overlapping rapidity-intervals around the centralrapidity(denoted by c r ) of the 32 S-AgBr(200 A GeV) interaction, has been analysed using the Visibility Graph method. There, it has been established that the multiplicity fluctuation in high energy interaction, follows scaling laws in pseudorapidity space.
In this experiment, we have have taken out the azimuthal(φ) counterpart for 4 of those 10 pseudorapidity(η)-space around c r , from the same interaction, to analyse the fluctuation and clustering pattern from complex network perspective and attempted to de-

Our method of analysis
tect whether the azimuthal fluctuation is also self-similar and follows scaling laws. Further the presence of collective anisotropic flow has been probed by extracting the clusters and calculating their Average-clustering-co-efficient, for each of the 4 φ-datasets, in the light of conventional multi-particle correlations.
The detailed steps for the analysis are described below.
1. For each dataset of η-values in 4 overlapping rapidityintervals around the c r (the range of the η-values is indicated by (c r − ∆η) to (c r + ∆η), where ∆η varies from 1 to 4), the corresponding φ-values are collected and this way the 4 input datasets for the experiment are formed. 4 Visibility Graphs are constructed as per the method described in Section. 2.1, for all these datasets.

Then clusters are extracted following Density-Based
Algorithm proposed by Ester et al. [36][Section 2.2] from each of the 4 Visibility Graphs and set of datapoints corresponding to each of the clusters, are obtained. Figure 2 shows a specimen set of 5 clusters taken out, from the dataset of φ-values for the corresponding the pseudorapidity region closest to c r for 32 S-AgBr(200 A GeV) interaction.
3. Then, again Visibility Graphs are constructed from all the cluster-datasets extracted from the 4 Visibility Graphs created in the Step 1. and for each graph the values of P (k)-s for all possible values of k-s are computed and for each P (k) versus k dataset, power-law fitting is done by following the method suggested by Clauset et. al. [45]. The power-law relationship has been confirmed from the corresponding χ 2 exp /DOF values and the values of R 2 . In the Fig 3-(a), the P (k) versus k plot for a sample cluster is shown, where the power-law relationship is evident from the value of R 2 of the power-law fitting. As explained in the Section 2.1, once the power-law has been confirmed for the P (k) versus k dataset, the power-law exponent is termed as the Power of the Scale-freeness in Visibility Graph or PSVG of the corresponding cluster.
The same values of PSVG parameter are obtained from the gradient of log 2 [P (k)] vs log 2 [1/k] plot of the same cluster-datasets. Again, good scaling behaviour is confirmed from the corresponding value of χ 2 exp /DOF and R 2 . Fig 3-(b) shows this for the same specimen cluster-dataset which is referred in the Fig 3-(a), where the PSVG is 1.83 ± 0.11. Similar analysis for all the clusters extracted from the Visibility Graphs constructed for all 4, φ-datasets around the c r of the experimental data, is carried out. As PSVG correlate with the amount of complexity and fractality of the data series and eventually the fractal dimension of the experimental data series [26,27,34], so it can be confirmed that all these clusters are scale-free and also of fractal structure.

Then Monte-Carlo simulated dataset is generated
for each of the cluster-datasets after assuming independent emission of pions in 32 S-Ag/Br interaction at 200A GeV. The data for Monte-Carlo simulated datasets have been chosen in such a way that dn dφ distribution of Monte-Carlo simulated data resembles the corresponding dn dφ of the real ensembles. Then for all these Monte-Carlo simulated datasets, first Visibility Graphs are constructed and then PSVG-s are calculated. This way, we finally obtain 4 sets of PSVG-values(calculated for both experimental and MC simulated cluster-datasets) for 4, φ-datasets. The PSVG-values calculated for experimental datasets significantly differ from their Monte-Carlo simulated counterpart. This finding establishes that the degree of complexity for any cluster is not the result of the Monte-Carlo simulated fluctuation pattern, but of the dynamics present in the clustering pattern.
5. In the next step, Average-clustering-co-efficients are extracted from the Visibility Graphs for all the cluster-datasets extracted from each of the 4, φdataset, as per the method explained in Section 2.2.
Then from the Visibility Graphs constructed from all the corresponding Monte-Carlo simulated datasets the Average-clustering-co-efficients are calculated. This way we obtain 4 sets of Average-clustering-coefficient(calculated for both experimental and MC simulated cluster-datasets) for the 4, φ-datasets.

Conclusion
In the Section 1, it has already discussed that azimuthal anisotropic distribution might be the underlying cause of collective anisotropic flow of the produced particles in ultra-relativistic heavy-ion collisions [5]. As the dynamics of collective anisotropic flow in multiparticle production process is yet to come out with accurate parameters and details and also, latest methods to analyse collective anisotropic flow using multi-particle azimuthal correlations, are not free from systematic bias in conventional differential flow analyses, hence this is still an open area of research. In view of this, we have attempted to analyse collective anisotropic flow by analysing the azimuthal distribution using a poles apart, rigorous approach of complex network perspective which gives more deterministic information about the anisotropy in azimuthal distribution by means of precise topological parameter. • It has already been shown that the multiplicity fluctuation in high energy interaction is self-similar and scale-free [1], where an exemplary data from 32 S-AgBr(200 A GeV) interaction has been analysed, using the Visibility Graph method and the PSVG values have been compared for all the overlapping η-regions around the c r .
In this work we have constructed Visibility Graphs from their corresponding azimuthal space or φdatasets. For each dataset cluster-datasets are extracted and then again Visibility Graphs are constructed for each cluster and its Monte-Carlo simulated counterpart. Finally it's shown that each of these clusters is self-similar, scale-free and hence of fractal structure.
Then for each cluster and its Monte-Carlo simulated counterpart, Average-clustering-co-efficients are calculated and it has been found that the count of clusters having almost similar Average-clusteringco-efficient to their Monte-Carlo simulated counterparts, is receding from the region closest to the farthest from c r .
• Clustering coefficients is essentially generalised to signed correlation between the nodes of the networks [46]. In this experiment the correlation is measured in terms of the visibility between the nodes of the cluster-datasets.
Hence this receding count of clusters in the Table 1 from rapidity-regions with ∆η = 1.0 to ∆η = 4.0 around c r , signifies that the particle to particle correlation in the azimuthal distribution is least in the closest rapidity-region around c r (with ∆η = 1.0) and hence this region has maximum number of clus-ters having similar Average-clustering-co-efficient to their Monte-Carlo simulated counterparts. This correlation is gradually increasing as the count of clusters having similar Average-clustering-co-efficient to their Monte-Carlo simulated counterparts decreases for the regions farther from c r and finally this count is least in the farthest rapidity-region from c r . Hence, particle to particle correlation in the azimuthal distribution is least in the central-most azimuthal space and gradually increasing to the one farthest from c r and becomes highest for the azimuthal distribution for the rapidity region with ∆η = 4.0.
• This in effect establishes the anisotropic property of azimuthal distribution in the central rapidity region which eventually might confirm the collective anisotropic flow in that region.