We show briefly the features of a percolation transition related to the networks obtained from a correlation matrix. The most interesting behaviour of this transition, investigated by numerical simulations with different thresholding rules, is that it shows a much faster transition from the disaggregated to the clustered phase, that resembles what has been described as an “explosive” percolation. A comparison with the “classic” random network percolation is shown, together with some applications of these concepts to the networks obtained from real data, that

behave differently depending on the data intrinsic structure

Remondini, Daniel

English

We show briefly the features of a percolation transition related to the networks obtained from a correlation matrix. The most interesting behaviour of this transition, investigated by numerical simulations with different thresholding rules, is that it shows a much faster transition from the disaggregated to the clustered phase, that resembles what has been described as an “explosive” percolation. A comparison with the “classic” random network percolation is shown, together with some applications of these concepts to the networks obtained from real data, that behave differently depending on the data intrinsic structure

REMONDINI, DANIEL

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Explosive percolation in correlation-based networks

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DOI 10.1393/ncc/i2011-10903-1Communications: SIF Congress 2010IL NUOVO CIMENTO Vol. 34 C, N. 5 Settembre-Ottobre 2011Explosive percolation in correlation-based networksD. Remondini(∗)Dipartimento di Fisica, Universita` di Bologna - Bologna 40127, Italy(ricevuto il 31 Dicembre 2010; approvato il 24 Marzo 2011; pubblicato online il 30 Giugno 2011)Summary. — We show briefly the features of a percolation transition related to thenetworks obtained from a correlation matrix. The most interesting behaviour of thistransition, investigated by numerical simulations with different thresholding rules,is that it shows a much faster transition from the disaggregated to the clusteredphase, that resembles what has been described as an “explosive” percolation. Acomparison with the “classic” random network percolation is shown, together withsome applications of these concepts to the networks obtained from real data, thatbehave differently depending on the data intrinsic structure.PACS 87.18.Vf – Systems biology.PACS 87.18.Wd – Genomics.PACS 64.60.aq – Networks.PACS 64.60.ah – Percolation.1. – DescriptionThe percolation transition in Erdo¨s-Re´nyi (E-R) random networks is a long-standingexample of phase transitions in networks [1]. Starting from a set of disjointed nodes,links are added randomly, and in the proximity of an average connectivity of one linkper node, a transition is observed for which a unique connected component develops thatoccupies a finite fraction of nodes in the thermodynamic limit.Very recently it has been shown that a slight modification of the growth rule can leadto a completely different behaviour at the transition point [2]. This transition has beencalled explosive, since it seems to produce a finite connected component all of a sudden,changing thus the transition properties. This discontinuity has been debated [3], but atleast it can be stated that anomalous critical exponents occur at such a transition.Network approaches have spread over several fields, from sociology to economics [4]and biology [5-7] to cite only some examples. In particular, in the study of times seriesdata, approaches based on the correlation matrices have been applied [4,8], that seem to(∗) E-mail: daniel.remondini@unibo.itc© Societa` Italiana di Fisica 203204 D. REMONDINIprovide robust results even when data are very noisy and the problem is ill-posed (whenthere are many more observed variables than time points available). This is a typicalsituation when dealing with recently available gene expression data, that can span thewhole genome (103–104 nodes) but for which very few time points are available (101 is atypical order of magnitude [9, 8]).We address the problem of percolation transition in the network obtained from correla-tion matrices, by thresholding of the correlation coefficients. The thresholding proceduremay be useful for removing noise from the data, but a problem is that a predefined thresh-old cannot be defined, if not based on single-node characteristics (statistical significanceof the correlation). Starting from randomly generated data, we observe an anomaloustransition with features similar to the explosive percolation, thus providing more infor-mation about the “null model” to be compared with real data. Moreover, applying thesame method to real data, we get some relevant information about the data structure,that seem to reflect features like modularity and non-random correlations.2. – Percolation by thresholdingThe “null model” is developed from a random Gaussian matrix X, (N M -dimensionalvectors) defined asX =⎛⎝x1. . .xN⎞⎠ ,(1)with xi ∝ N(0, σ).Hence we define the N ×N correlation matrix C as usual:cij =E[(xi − μi) · (xj − μj)]σi · σj .(2)The correlation matrix gives us the chance to build a network in three different ways,according to the thresholding (CT > 0) we choose– correlation: link if cij ≥ CT ,– anticorrelation: link if cij ≤ −CT ,– both: link if c2ij ≥ CT .Taking different threshold values we get networks with a different number of links, inthis way we follow the evolution of our networks just modifying the threshold value. Westart with N isolated nodes and we finally reach a fully connected network.The network dynamics is reflected by the formation of a giant component. We com-pared our models with the classical percolation problem and with a peculiar explosivepercolation model due to a modification of the growth rule, the so-called Product Rule(PR [2]).We have observed that, for the correlation-based percolation, the properties depend onthe number of dimensions of the vectors (M), since for high-dimensional vectors (M  1)the Gaussian limit for the distribution of cij is recovered, and the percolation becomesindistinguishable from the classical one. We fixed M = 5 so to appreciate the differencesEXPLOSIVE PERCOLATION IN CORRELATION-BASED NETWORKS 2050 1 2 3 4 5 6 7 8 9 1000.10.20.30.40.50.60.70.80.91GC %<K>  anticorrE−RPRFig. 1. – Percolation for different models with N = 20000 and M = 5: Giant Component sizefor Erdo¨s-Re´nyi, Correlation, Anticorrelation, Product Rule models.between the correlation coefficients distribution and a normal Gaussian distribution, andalso in accordance with the size of the real-data set vectors as shown below.As shown if fig. 1, E-R percolation is the less steep, and PR is the steepest. Giantcluster percolations for the correlation and the anticorrelation cases are less steep thanPR, but start later as compared to E-R, thus they have a shorter “latent phase” betweengiant cluster onset and its occupation of the whole network.3. – Example with real dataIn previous papers we studied the dynamics of a gene expression time series net-work [8]; we compared two data sets of gene expression obtained from a set of microarrayexperiments using genetically engineered rat fibroblast cell lines, in which a master genec-myc was, respectively, silenced (producing the N data set) and overexpressed (the Tdata set). We had 5 time points for both data sets (thus M = 5 according to the notationintroduced before) and 8799 probes for each array.In the correlation-based model, the similarity measure for the expression dynamics oftwo genes is given by the correlation between the two expression-level time series. For agiven data set, if we define xlj as the expression level of a gene with label l at time j,then the similarity between two genes with labels l and r, respectively, is given byclr =∑j(xlj − μl)(xrj − μr)σlσr,(3)where μl and μr are the averages in time of the expression levels for the two genes, andσl and are σr their standard deviations.The correlation approach can be motivated by the hypothesis that genes belongingto the same activation (or inhibition) pathway should present similar (or opposite) ex-pression profiles in time.Hence, once we have the correlation matrix the adjacency matrix is obtained byconsidering thresholding on c2ij , as we explained previously.206 D. REMONDINI0.98 0.985 0.99 0.995 100.10.20.30.40.50.60.70.80.91r2GC %  TshufTNFig. 2. – Percolation for different data sets: N , T and T reshuffled, whole array with 8799 probes.Plot of Giant Cluster (GC) relative size (with respect to the whole network) as a function of r2correlation value.We checked both the whole 8799 probe set, and a subset of 1191 probes obtainedby statistical analysis over N and T cases (2-way ANOVA, as explained in [8]). Asa comparison, we randomly reshuffled data in both data sets, and performed the sameanalysis as for the unshuffled data (only the T reshuffled data set is shown, but both Nand T produce similar results, thus demonstrating that reshuffling destroys any informa-tion contained in the time series data).As shown in figs. 2 and 3, we observe very different percolation dynamics between thethree data sets: high values of correlation last longer for the real data, while reshuffleddata resemble percolation with randomly generated data (not shown). Moreover, whereas0,95 0,96 0,97 0,98 0,99 100.10.20.30.40.50.60.70.80.91r2GC %  NTTshufFig. 3. – Percolation for different data sets: N , T and T reshuffled, selection of 1191 significantprobes. Plot of Giant Cluster (GC) relative size (with respect to the whole network) as afunction of r2 correlation value.EXPLOSIVE PERCOLATION IN CORRELATION-BASED NETWORKS 207the shuffled data giant component grows more continuously, in real data we observe somediscontinuous “jumps” in its size (see, in particular, fig. 3, T data set for r2 values closeto 0.99). This discontinuities may resemble the formation of superparamagnetic domainsduring cooling [10], thus it should be interesting to further investigate the cluster sizeand distribution closer to these values in order to search for modules inside the wholenetwork.4. – ConclusionsWe have observed percolation dynamics in the growth of a network based on thecorrelation of randomly generated data, that seems to differ from classical percolation,and results to be more similar to the so-called explosive (or better anomalous) percolationmodels. We investigated the behaviour of real biological data (high-throughput geneexpression time series), obtained in an experiment in which a very important biologicalmechanism is switched on and off. We observe striking differences in percolation dynamicsalso in this case, both at the level of global gene expression level, and in a subset of genesselected by their significant response to the switching. Some hints point out at a possiblerole of percolation dynamics in finding modules and structures inside such networks, thatdeserves further investigation.∗ ∗ ∗DR acknowledges Progetto Strategico d’Ateneo 2006 “p53 e patologie non neoplastichenell’anziano: uno studio multidisciplinare sul ruolo del polimorfismo al codone 72 delgene TP53”, Bologna University. DR also would like to thank G. Menichetti for herprecious contributions to the development of this research project.REFERENCES[1] Erdo¨s P. and Re´nyi A., Publ. Math. Inst. Hung. Acad. Sci., 5 (1960) 17.[2] Achilioptas D. et al., Science, 323 (2009) 1453.[3] da Costa R. A. et al., Phys. Rev. Lett., 105 (2010) 255701.[4] Tumminello M. et al., J. Econ. Behav. Organ., 75 (2010) 40; doi:10.1016/j.jebo.2010.01.004.[5] Wuchty S. et al., The Architecture of Complex Networks, in Complex Systems Science inBiomedicine, edited by Deisboeck T. S. and Yasha Kresh J. (Springer) 2006, pp. 165-181.[6] Milo R. et al., Science, 298 (2002) 824.[7] Tieri P. et al., Bioinformatics, 21 (2005) 1639.[8] Remondini D. et al., Proc. Natl. Acad. Sci. U.S.A., 102 (2005) 6902.[9] Iyer V. R. et al., Science, 283 (1999) 83.[10] Domany E., Physica A, 263 (1999) 158.

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Explosive percolation in correlation-based networks

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