Supplementary Materials for A universal data based method for reconstructing complex networks with binary-state dynamics

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1 Supplementary Materials for A universal ata ase metho for reonstruting omplex networks with inary-state ynamis Jingwen Li, Zhesi Shen, Wen-Xu Wang, Celso Greogi, an Ying-Cheng Lai 1 Computation etails Parameter values ihe inary-state ynamis use for network reonstrution are isplaye in Supplementary Tale S1. The only requirement for hoosing the parameter values is that the swithing ynamis shoul e monotoni. Sine all the inary-state ynamis are monotoni, there is no speifi restrition for the parameter values. Note that several moels have onvergent ehaviors. If the states of noes onverge into a stale state, there will e no more useful information for network reonstrution. If this ours, we ranomly initialize the states of all noes after a ertain perio. The set of the threshol parameter for realizing the merging proess for network reonstrution is inepenent of network struture an inary-state ynamis. We investigate the epenene of the reonstrution performane ohreshol. The results are shown in Supplementary Fig S1. We fouhat an an always reah high values when.4.55 in all ases. Thus, we set the threshol to e.45 for simpliity. Regaring the seletion of ases, the metho is relatively time onsuming eause it requires alulating the Hamming istane etween eah pair of strings in ifferent time steps. Hene, to improve omputational effiieny, for large-size networks with N 5, we hoose ases ranomly instea of using the ase-seletion metho presente ihe maiext. It reues auray a little in a few ases, ut the omputational omplexity is onsieraly reue. Supplementary Figs. S2 (a) an () show the results of reonstrution for Ising an ynamis on, WS 1

2 Supplementary Tale S1 Settings in numerial simulations. Parameter values in various inary-state ynamis ahe perio for initiating noe states eause of onverging to steay state. Moel Parameters Convergent Upate perio Yes 1 (5 for N=1) Kirman 1 =.1, 2 =.1, =.8 No Ising Gluaer β = 2 No SIS λ =.2, µ =.5 No α =.1, β = 1, a = 6, = 5, = 1, = No Language s =.5, α =.7 No Threshol M k = 2/k Yes 5 vote Q =.3 Yes 1 (5 for N=1) 2

3 an networks. We fouhat for Ising ynamis, the results are almost not affete y the value of σ; ut for ynamis, a large value of σ is preferre. The possile reason is that, for ynamis with onvergene suh as, the time series is ominate y all zeros or all ones. Ihe similarity networks, the noes representing all zeros(or ones) time strings ensely onnet to eah other, whih leas to a imoal egree istriution, as shown in Supplementary Fig. S2(). We an see that there are more than 4% noes ihe rightmost in. Thus, σ shoul e large enough to exlue these noes, ahehe reonstruting performane will approah high auray then, as shown in Fig. S2(). For Ising ynami, the egree istriution is like a ell shape, ahere are no ominant zeros(or ones) time strings, so σ is not a key parameter. We also ompare the performane of the networke ase seletion with the performane of ranomly ase seletion. The networke inee shows etter performane, espeially for ynamis with onvergene. a ,,, Ising, Ising, Ising, ,,, Ising, Ising, Ising, Supplementary Figure S1 Determination of threshol. (a) as a funtion of threshol parameter for the voter an Ising moel on, an networks. () as a funtion of for the two moels ahree networks. The network size N = 1 an k = 6. The length of time series is Other parameters of ynamis are shown in Supplementary Tale S1. 3

4 a A UROC egree istriution ising ran-sw net-sw ran-sf net-sf ran-er net-er σ voter σ noe egree noe egree Supplementary Figure S2 Determination of threshol σ. (a,) as a funtion of threshol parameter σ for (a) the voter an () Ising moel on, an networks, respetively. The ashe lines are the results of ranomly selete ases. () The egree istriution of the onstrute similarity network for Ising ynami on network. () The egree istriution of the onstrute similarity network for ynami on network. The network size N = 1 an k = 6. The length of time series is Other parameters of ynamis are shown in Supplementary Tale S1. There is an ajustale parameter λ ihe lasso. In general, the parameter is etermine y using ross-valiation metho, suh as sklearn.linear moel.lassocv in python. Ierms of the ross-valiation metho, we otaine the proper value of λ, whih is set to e 1 4 an 1 3 for reonstruting networks with N 5 an N = 1, respetively, in all reonstrutions. All the onvex optimizations are implemente in Python(version 2.7) an Sklearn(version.14). 4

5 2 Depenene of performane on ata amount We examine how the length of time series affets reonstrution auray. We let enote the ratio of the total length of time series normalize to the network size N. Supplementary Fig. S3 shows the reonstrution performane measure y an for various ynamis in omination with ifferent types of networks. We fihat an rapily inreases as inreases. After exees a relatively small value, nearly full reonstrution an e ahieve, whih provies aitional eviene for the high effiieny of our metho. The results are summarize in Tale II ihe maiext. a Kirman Ising SIS e f Language g Threshol h i j Kirman k Ising l SIS m n Language o Threshol p Supplementary Figure S3 Reonstrution performane with respet to the length of time series. (a-h) an (i-p) as funtions of the normalize length of time series for various ynamis on, an networks. The network size N = 5 an k = 6. Other parameter values of inary-state ynamis are shown in Supplementary Tale S1. 5

6 3 Roustness against noise an missing ata Roustness against noise an missing ata is important for evaluating the appliaility of a metho. We onsier the senario of noise-inue wrong reors iime series. Speifially, we assume that a fration n f of inary states are wrong, an flip from 1 to zero or from zero to 1. The presene of unoservale noes or missing ata is quite often ihe real situation. We assume that the ata of a fration of noes, n m, annot e oserve. We investigate the reonstrution auray as a funtion of n f an n m, respetively. As shown in Supplementary Fig. S4 an Supplementary Fig. S5, respetively, we fihat high an remains in a wie range of n f an n m, proviing strong eviene for the roustness of our reonstrution framework against measurement noise an missing ata. The results are summarize in Tale III ihe maiext. a e f n f n f n f Supplementary Figure S4 Roustness against measurement noise. (a,,) an (,e,f) as funtions of the fration n f of wrong states iime series for the voter, Ising an majority moel on, an networks. Parameters of networks an ynamis are the same as in Supplementary Fig. S3. = 1. 6

7 a e f n m n m n m Supplementary Figure S5 Roustness against missing ata. (a,,) an (,e,f) as funtions of the fration n m of unoservale noes for the voter, Ising an majority moel on, an networks. Parameters of networks an ynamis are the same as in Supplementary Fig. S3. = 1. 7