We present a generic threshold model for the co-evolution of the structure of
a network and the state of its nodes. We focus on regular directed networks and
derive equations for the evolution of the system toward its absorbing state. It
is shown that the system displays a transition from a connected phase to a
fragmented phase that depends on its initial configuration. Computer
simulations are performed and confirm the theoretical predictions.Comment: 4 pages, 4 figure

Gonzalez-Avella, Juan Carlos

Lambiotte, Renaud

English

arXiv

We present a generic threshold model for the co-evolution of the structure of a network and the state of its nodes. We focus on regular directed networks and derive equations for the evolution of the system toward its absorbing state. It is shown that the system displays a transition from a connected phase to a fragmented phase that depends on its initial configuration. Computer simulations are performed and confirm the theoretical predictions

Gonzalez-Avella, JC

ORA - Oxford University Research Archive

Coeovolutionary threshold dynamics

4 pages, 3 figures.-- arXiv:0906.3389v1, submitted on 18 Jun 2009.--  PACS numbers: 89.75.Fb, 87.23.Ge, 05.90.+mWe present a generic threshold model for the co-evolution of the structure of a network and the state of its nodes. We focus on regular directed networks and derive equations for the evolution of the system toward its absorbing state. It is shown that the system displays a transition from a connected phase to a fragmented phase that depends on its initial configuration. Computer simulations are performed and confirm the theoretical predictions.J.C. G-A. acknowledges support from MEC (Spain)
through the project FISICOS FIS2007-60327. R.L. has
been supported by UK EPSRC. This work was conducted
within the framework of COST Action MP0801 Physics
of Competition and Conflicts.Peer reviewe

González-Avella, Juan Carlos

Digital.CSIC

arXiv:0906.3389v1  [physics.soc-ph]  18 Jun 2009Coeovolutionary Threshold DynamicsRenaud Lambiotte1 and Juan Carlos Gonza´lez-Avella21 Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, SW7 2PG London, UK2 IFISC, Instituto de F´ısica Interdisciplinary Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca, SpainWe present a generic threshold model for the co-evolution of the structure of a network and thestate of its nodes. We focus on regular directed networks and derive equations for the evolution of thesystem toward its absorbing state. It is shown that the system displays a transition from a connectedphase to a fragmented phase that depends on its initial conﬁguration. Computer simulations areperformed and conﬁrm the theoretical predictions.PACS numbers: 89.75.Fb, 87.23.Ge, 05.90.+mThe relation between a cause and its effect is usuallyabrupt in complex systems, in the sense that a smallchange in the neighborhood of a subsystem may (or not)trigger its reaction. This mechanism is at the heartof many models of self-organized criticality [1] where acascade starts when the system has been frustrated be-yond some threshold, e.g. the angle of a sand pile, butalso in models for the diffusion of ideas in social net-works [2, 3, 4, 5] where the adoption of a new idearequires simultaneous exposure to multiple active ac-quaintances, and in integrate-and-fire neuron dynam-ics [6] where the voltage on a single neuron increasesuntil a specified threshold is reached and it suddenlyfires by emitting an action potential, thereby quicklyreturning to its reference. These types of model con-sist in cascading propagations on a fixed topology, i.e.,a network of some sort, until a frozen configuration isreached, but they do not incorporate the well-knownfeed-back existing between network topology and dynam-ics [7, 8, 9, 10, 11, 12, 13, 14, 15], namely that the topol-ogy itself may reorganize when it is not compatible withthe state of the nodes. This reorganization may originatefrom homophily and social balance in social networks orsynaptic plasticity in neuron dynamics.The purpose of this paper is to introduce a model forcoevolutionary threshold dynamics (CTD). Let us de-scribe its ingredients in terms of diffusion of opinions insocial networks [16, 17] while keeping in mind that themodel is applicable to more general systems. The systemis made of a social network of interaction, whose N nodesare endowed with a binary opinion s, + or −. The dy-namics is driven by the threshold φ, such that 0 ≤ φ ≤ 1and, in most cases of interest, φ > 1/2. At each step,a randomly selected node i evaluates the opinion of itski neighbours. Let φi be the fraction of neighbours dis-agreeing with i. If φi ≤ φ, node i breaks the links towardthose disagreeing neighbours and rewires them to ran-domly selected nodes. If φi > φ, i adopts the state of themajority. By construction, the dynamics perdures untilconsensus, i.e., all agents having the same opinion, hasbeen attainted in the whole system or in disconnectedcomponents. This absorbing state obviously depends onthe threshold φ but also, as we will discuss below, on itsinitial condition.FIG. 1: Update process of CTD for two diﬀerent conﬁgura-tions of neighbors. When one out of four neighbors is in adiﬀerent state, the central node breaks its links and creates anew link to a randomly chosen node. When three out of fourneighbors are in a diﬀerent state, the threshold φ is exceededand the central node thus adopts the majority state.A complete analysis of CTD requires extensive com-puter simulations, which is not the objective of this pa-per. We will instead focus on a simplified version of themodel that can be studied analytically and pinpoint thekey mechanisms responsible for its behaviour [27]. Inthis simplified version, the network is directed and allthe nodes have two incoming links, i.e. each is influencedby exactly two nodes, while their out-degree is initiallyPoisson distributed. Moreover, we will take φ = 1/2,such that CTD now simplifies as follows. At each timestep, a node i is selected at random. If i is surrounded bytwo nodes with different opinions, it switches its opinion,i.e., si → −si. If the opinion of only one of its neighbours,say j is different, i cuts its link from j and reconnects toa randomly chosen node, i.e., its in-degree remains con-stant. It is interesting to stress that the choice φ = 1/2for a directed network with a constant in-degree of twocorresponds to the unanimity rule [18] when no rewiringis implemented. This model is well-known to exhibit anon-trivial relation between initial and final densities of2+ nodes, denoted by n+;0 and n+;∞ respectively. Wewill show that the addition of the rewiring mechanismleads to a transition from a connected phase with con-sensus where all the nodes asymptotically belong to thesame cluster, to a fragmented phase where two discon-nected clusters of different opinions survive. The criticalparameter of this transition is shown to be the initialdensity n+;0 of + nodes, i.e.,n+;0 < nc or n+;0 > 1− nc one single component,nc < n+;0 < 1− nc two disconnected components,(1)where nc is the critical density.In order to analyze the system dynamics, let us followthe approach proposed in [18] and focus on the numberNs0;s1s2 of configurations where a node in state s0 re-ceives its incoming links from a node in state s1 and an-other node in state s2. Let us denote by {s0; s1s2} such atriplet of nodes. By construction, si may be +1 or−1 and∑s0s1s2Ns0;s1s2 = N . Moreover, the order of the linksis not important and therefore Ns0;s1s2 = Ns0;s2s1 . Byneglecting higher order correlations than those includedin Ns0;s1s2 , it is possible to derive the set of equationsN+;++(t+ 1) = N+;++ +1N(N−;++ + n+N+;+− + pi−→+N+;+− − 2pi+→−N+;++)N+;−−(t+ 1) = N+;−− +1N(−N+;−− + pi+→−N+;+− − 2pi−→+N+;−−)N+;+−(t+ 1) = N+;+− +1N(−n+N+;+− + 2pi−→+N+;−− + 2pi+→−N+;++ − (pi+→− + pi−→+)N+;+−)N−;−−(t+ 1) = N−;−− +1N(N+;−− + n−N−;+− + pi+→−N−;+− − 2pi−→+N−;−−)N−;++(t+ 1) = N−;++ +1N(−N−;++ + pi−→+N−;+− − 2pi+→−N−;++)N−;+−(t+ 1) = N−;+− +1N(−n−N−;+− + 2pi+→−N−;++ + 2pi−→+N−;−− − (pi+→− + pi−→+)N−;+−) (2)where n+ and n− are the density of + and − links respec-tively. pi+→− (pi−→+) is the probability for a randomlyselected + (−) node to switch its opinion to - (+). Byconstruction, this quantity is the probability that a ran-dom + (−) node is connected to two − (+) nodespi+→− =N+;−−N+, pi−→+ =N−;++N−, (3)where N+ =∑s1,s2N+;s1s2 and N− = N − N+ are thetotal number of + and − nodes respectively.Let us describe in detail the first equation for N+;++,the other ones being obtained in a similar way. Its evo-lution is made of several contributions. The first termis the probability that a {−; ++} triplet is selected andtransforms into {+;++} by unanimity rule. The secondterm is the probability that a {+;+−} triplet is selectedand the rewired link (originally from + to −) arrives ona + node (with probability n+). The last two terms ac-count for possible change of the state of one of the twoneighbours in the triplet, as they may also switch theiropinion because of a unanimity rule in another triplet,and are evaluated by using the aforementioned pi+→−and pi−→+.As discussed in [18], several initial conditions may inprinciple be chosen for the system of equations (2), eachof them leading to its own trajectory in the 6-dimensionaldynamical space. Such initial conditions are subject tothe normalization∑s0,s1,s2Ns0;s1s2 = N , and to the con-servation lawsT+ = 2N+, T− = 2N−, (4)where the quantitiesT+ = 2N+;++ + 2N−;++ +N+;+− +N−;+−T− = 2N+;−− + 2N−;−− +N+;+− +N−;+− (5)are the total number of + (−) incoming neighbors inthe triplets. Relations (4) simply mean that each nodei that is a neighbor in a triplet {sx; sisy} is also at thesummit of another triplet {si; sx′ sy′} (as it also receivestwo incoming links by construction). In order to selectone of the several configurations Ns0;s1s2 that still satisfythe above constraints, we will further assume that theinitial configuration is uncorrelated and therefore thateach node has the same probability n+;0 to be +. Amongall the possible configurations for which N+ = Nn+;0, wetherefore select the initial conditionN+;++ = Nn3+;0N+;−− = Nn+;0(1− n+;0)2N+;+− = 2Nn2+;0(1 − n+;0)N−;−− = N(1− n+;0)3N−;++ = N(1− n+;0)n2+;0N−;+− = 2N(1− n+;0)2n+;0. (6)3FIG. 2: In the upper ﬁgure, we plot the relation (8) betweenthe initial density and the ﬁnal density of + nodes, evaluatedby integrating the set of Eqs.(2) and by performing numericalsimulations of a network made of 1000 nodes. We also plotthe order parameter r = 〈1/2− |1/2−n+,∞|〉 which conﬁrmsthat the system actually breaks into disconnected componentswhen nc < n+,0 < 1−nc, i.e., we only plot (8) in the interval[0, 0.5] as the curves are symmetric around the point (0.5, 0.5).In the lower ﬁgure, we plot the probability density ρ(n+;∞)that the absorbing state has a density n+;∞ of + nodes forN = 100 and N = 1000 when n+;0 = 0.4.Before going further, it is instructive to look at thetotal number N+ =∑s1,s2N+;s1s2 of + nodes whosetime evolution is obtained by summing over the first threeequations of (2)N+(t+ 1) = N+ +1N(N−;++ −N+;−−). (7)This relation shows that N−;++ = N+;−− at stationarity.A careful look at the second equation of (2) shows, how-ever, that N+;−− has to decay until it reaches zero. Thethird equation of (2) also shows that the only stationarysolution ofN+;+− is also zero when N−;++ = N+;−− = 0,thereby confirming that the dynamics asymptoticallyreaches a frozen configuration where consensus is reachedamong connected nodes. The dynamics is thereforedriven by two types of triplets: the triplets {+;−−} and{−; ++} drive the system toward consensus, while theconfigurations {+;+−} and {−; +−} allow for a topo-logical rearrangement of the network. This rearrange-FIG. 3: Visualisation of the initial and ﬁnal states of one real-ization of the dynamics for a network made of N = 100 nodes.The initial density of + nodes n+,0 = 0.4. The asymptoticnetwork is made of two clusters. The + cluster is now madeof 35% of the nodes.ment implies that the only frozen states are those cor-responding to consensus (in one or several clusters) [28]and drives the division of the system into disconnectedclusters. The competition between these two types ofmechanisms is crucial for the transition (1). One shouldalso note that models for opinion dynamics are knownto exhibit coexistence of different opinions when appliedto a static underlying network with modular structure[20]. In the case of CTD, in contrast, it is the rewiring ofthe links that reorganizes the system into modules andthereby allows for coexistence.By integrating recursively the system of Eqs.(2) start-ing from the initial conditions (6), we obtain a non-trivialrelationn+,∞(n+,0) (8)between the initial density and the final density of +nodes. This numerical integration confirms the abovediscussions, and clearly shows that a transition occursat nc ≈ 0.22 (see Fig. 2). One should insist on the factthat this relation differs from the standard exit probabil-ity measured when the dynamics takes place on a staticnetwork [21, 22, 23]. In the latter models, n+,∞(n+,0)would measure the probability to end in a + consensus(in the whole system) starting from some initial densityof + nodes. Relation (8) is more reminiscent of the stan-dard unanimity rule [18] without rewiring, where the sys-tem asymptotically reaches a frozen state different fromconsensus at each realization, and where n+,∞ is the av-erage number of + nodes in this frozen state. Because4of the rewiring process, however, a non-vanishing valueof n+,∞ also implies that the system has split into twodisconnected clusters and that a different consensus hasbeen reached in each cluster.We have verified the accuracy of our calculations byperforming numerical simulations of the model (see Fig.2). To do so, we have considered systems made of 1000nodes and have averaged the asymptotic density of +nodes (evaluated when the dynamics is frozen) over 1000realizations for each value of n+,0. In order to checkthat the system actually breaks into two clusters whennc < n+,0 < 1 − nc (see Fig. 3), we have also measuredr = 〈1/2 − |1/2 − n+,∞|〉. This order parameter wouldvanish if, for each realization, n+,∞ is either zero or one,while r = 〈n+,∞〉 if the system breaks into two clus-ters. The simulation results show an excellent agreementwith the theoretical predictions and confirm the fragmen-tation of the network at the critical value nc. Finally,we have also looked at the probability density ρ(n+,∞),i.e.,∫ 10ρ(x)dx = 1, that the absorbing state is made ofn+,∞N nodes. This quantity is measured by performing5× 104 simulations starting from the same initial condi-tion n+;0 = 0.4. The distribution is shown to be peakedaround its average, in contrast with the two delta peaksat 0 and 1 that would be expected if full consensus hadbeen the only absorbing state.To conclude, we have focused on a model for coevolu-tionary threshold dynamics where the binary state of anode and its links coevolve. We have shown that the sys-tem may undergo fragmentation, a feature that has beenobserved in other coevolution network models, based onthe Axelrod [12] or the Voter model [15] for instance, butalso in the case of coupled maps with variable couplingstrength [24]. In the case of CTD, the critical parameteris the initial condition, as a sufficient fraction of + nodesis necessary for such nodes to survive and to separatefrom the main cluster. In this paper, we have focused ona simplified version of CTD where the underlying net-work is directed and regular, and where the in-degreehas the smallest non-trivial value, i.e., two. Additionalcomputer simulations are therefore required in order toexplore the role of the threshold φ on the asymptoticstate in more complex directed or undirected networks.Finally, let us point to an interesting generalization ofCTD that would include two different threshold φr andφa for either rewiring links from disagreeing neighboursor adopting the state of the majority. Such a modelwould unify two seminal threshold models, namely theGranovetter model for the diffusion of cultural traits [2]and the Schelling model for social segregation [25].AcknowledgementsJ.C. G-A. acknowledges support from MEC (Spain)through the project FISICOS FIS2007-60327. R.L. hasbeen supported by UK EPSRC. This work was conductedwithin the framework of COST Action MP0801 Physicsof Competition and Conflicts.Added Note: After this manuscript was completed, webecame aware of a related work by Mandra et al. [26].[1] H.J. Jensen, Self-Organized Criticality (Cambridge Uni-versity Press, Cambridge, England, 1998).[2] M. Granovetter, Am. J. Social. 83, 1420 (1978).[3] D. J. Watts, Proc. Natl. Acad. Sci. USA 99, 5766 (2002).[4] J. Kleinberg, Cascading Behavior in Networks: Algorith-mic and Economic Issues. In Algorithmic Game The-ory (Cambridge University Press, Cambridge, England,2007).[5] D. Centola, V. M. Egu´ıluz, M. W. Macy, Physica A 374,449456 (2007).[6] A.V. Holden, Models of the stochastic activity of neurons(Springer-Verlag, Berlin, Germany, 1976).[7] M.G. Zimmermann, V.M. Egu´ıluz, and M. San Miguel,Phys. Rev. E 69, 065102(R) (2004).[8] G.C.M.A. Ehrhardt, M. Marsili, and F. Vega-Redondo,Phys. Rev. E 74, 036106 (2006).[9] S. Gil and D.H. Zanette, Phys. Lett. A 356, 89 (2006).[10] P. Holme and M.E.J. Newman, Phys. Rev. E 74, 056108(2006).[11] D. Garlaschelli, A. Capocci and G. Caldarelli, NaturePhysics 3, 813-817 (2007).[12] D. Centola, J.C. Gonza´lez-Avella, V.M. Egu´ıluz, and M.San Miguel, J. of Conflict Resol. 51, 905 (2007).[13] C. Nardini, B. Kozma, and A. Barrat, Phys. Rev. Lett.100, 158701 (2008).[14] I. J. Benczik, S. Z. Benczik, B. Schmittmann, and R. K.P. Zia, Europhys. Lett. 82, 48006 (2008).[15] F. Vazquez, V.M. Eguiluz, M. San Miguel, Phys. Rev.Lett. 100, 108702 (2008).[16] S. Galam, Journal of Mathematical Psychology 30, 426-434 (1986).[17] C. Castellano, S. Fortunato and V. Loreto, Reviews ofModern Physics 81, 591-646 (2009).[18] R. Lambiotte, S. Thurner and R. Hanel, Phys. Rev. E76, 046101 (2007).[19] P. Klimek, R. Lambiotte and S. Thurner, Europhys. Lett.82, 28008 (2008).[20] R. Lambiotte, M. Ausloos and J. Ho lyst, Phys. Rev. E75, 030101(R) (2007).[21] P. L. Krapivsky and S. Redner, Phys. Rev. Lett. 90,238701 (2003).[22] M. Mobilia and S. Redner, Phys. Rev. E 68, 046106(2003).[23] R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007(2008).[24] J. Ito and K, Kaneko, Phys. Rev. E 67, 046226 (2003).[25] T.C. Schelling, J. of Math. Soc. 1, 143-186 (1971).[26] S. Mandra, S. Fortunato and C. Castellano,arXiv:0906.2471.[27] Preliminary simulations show that the transition pre-sented in this paper also takes place in the case of undi-rected networks and for diﬀerent values of φ.[28] This mechanism reminds of the ’social temperature’ de-ﬁned in [19] where random rearrangements of the linksdrive the system toward consensus.

Coeovolutionary Threshold Dynamics

Abstract

Similar works

Full text

Available Versions

ORA - Oxford University Research Archive

Digital.CSIC

Oxford University Research Archive

Oxford University Research Archive (ORA)