connectivity is key to understanding activity in neural systems ( sporns et al . , 2000 ) .
network connectivity in science and in engineering fields as diverse as mechanics , communication technology , public health , geography and town planning , is studied mathematically using the concepts of graph theory ( bollobas , 1998 ) .
recently , graph theory is being applied to brain connectivity ( sporns and zwi , 2004 ; bullmore and sporns , 2009 ) and its pathologies in alzheimer 's disease ( stam , 2004 ; stam et al . , 2007 ) , brain tumors ( bartolomei et al . , 2006 ) , epilepsy ( ponten et al . , 2007 ) and , in particular , schizophrenia ( bleuler , 1911/1950 ; friston and frith , 1995 ; andreasen , 1999 ; micheloyannis et al . , 2006 ;
applying graph - theoretic concepts to the brain sheds new light on the basic principles of integration and segregation underlying adaptive cognitive processes , and on their disruption in maladaptive states .
schizophrenia has been understood as a cognitive disorder ( bleuler , 1911/1950 ) based on the breakdown of large - scale cortico - cerebellar - thalamic - cortical ( andreasen , 1999 ) or prefronto - temporal circuits ( friston and frith , 1995 ; goldman - rakic and selemon , 1997 ) , or more generally the inability to integrate neural processes in different brain areas , a syndrome termed dysconnectivity ( stephan et al .
this condition may pare down , in particular , the input to pyramidal cells of the dorsolateral prefrontal and temporal cortex ( garey et al .
these cells are glutamatergic and receive projections from the thalamus and widespread cortical areas , and hence are likely to be involved in higher - level cognition .
reduced connectivity may thus lead to fragmentation , a loss of coherence in cognitive activity .
graph theory enables us to model the loss of connectivity in simulated neuronal networks and predict the time course of fragmentation . on the face of it ,
percolation plays an important role in the evolution , growth , and maintenance of a large variety of natural , technological , and social systems ( ben avraham and havlin , 2000 ) .
it refers to the probability of existence of a path between every pair of nodes in a graph , or equivalently , the graph being  connected 
whilst many previous studies have examined cortical network connectivity in schizophrenia and other disorders , none to our knowledge have employed the concept of percolation , an issue that we presently redress .
the percolation function is the cumulative density function ( cdf ) of percolation as a function of connectivity .
it is possible , in principle , to measure the percolation function in living neural tissue , by using progressive lesioning , for instance through the administration of inhibitory neurotransmitters ( breskin et al . ,
observations on human brain functional connectivity may be compared to the theoretical percolation function for random networks ( erds and rnyi , 1959 ) .
the percolation function cp(n ) of a random graph of n vertices and e edges is given by cp(n ) = e the presence of a critical threshold motivates us to revisit the notion of cortical dysconnectivity as a sudden breakdown of percolation .
there are , however , reasons to assume that the percolation threshold is neither the first , nor the most predominant , critical transition in the development of schizophrenia : brains are not random networks . in both the structural ( sporns and zwi , 2004 ) and functional ( salvador et al .
2006 ) domains , the hallmarks of brain organization include local clustering as expressed in high values for the clustering coefficient ( cc ) , high global connectedness as specified by a short characteristic path length ( cpl ) ( watts and strogatz , 1998 ) , and modularity ( murre and sturdy , 1995)a combination characteristic of modular small - world networks ( he et al . , 2009 ) .
graph - theoretical studies ( murre and sturdy , 1995 ; watts and strogatz , 1998 ) showed that small - world and modular networks can secure global connectivity with a small number of connections . for brains configured as modular small - worlds , a few connections will suffice to ensure percolation .
most likely , therefore , percolation is not the crucial bottleneck for brain pathologies such as schizophrenia .
we will propose as an alternative theoretical possibility that , instead , brain pathologies are associated with a breakdown in the local organization . in schizophrenia patients ,
functional connectivity in scalp eeg channels appears to reflect a loss of clustering after correcting for differences in the density of functional connections ( micheloyannis et al .
the question , therefore , arises , whether fragmentation can be understood as a critical breakdown in the ability of the brain to establish and maintain a modular small - world functional architecture .
here we show by numerical simulations that in neural activity networks , with loss of connectivity a self - organizing small - world neural network can not sustain its local clustering , well before global connectivity breaks down .
the iterated logistic map f(x)= 1  ax is unimodal on [ 1;1 ]  [ 1;1 ] and capable of both periodic and chaotic behavior depending on its control parameter a. in this study , we construct networks of coupled logistic maps , all with parameter a = 1.7 such that the dynamics of a single unit are chaotic under iteration of a randomly chosen initial activation value . a unit x is coupled with coupling strength  = 0.4 to any number mi of other units in the network such that its activation value xn + 1 at iteration n + 1 depends on the activation value of itself and all adjacent units at iteration n : 

 in this equation , b(i ) is the set of units adjacent to unit i , mi is the number of units adjacent to unit i. the coupling strength  is divided by mi , and has a compensation term ( 1   ) to make sure that logistic map of an individual unit retains its mapping [ 1:1 ]  [ 1:1 ] , and thus functions properly for any numbers of adjacent units . a network of this type can be used to study the buildup and breakdown of modularity resulting from hebbian adaptive structural self - organization .
it implements a simple rewiring rule based on synchronization of chaotic activity and rewires at most one connection per iteration , carefully keeping the network 's total number of connections constant throughout the process .
a single iteration of the network consists of four steps , to be repeated several times after an initial random inception : 
 initialize the network .
randomly establish e connections between v units to create a ( v , e ) random network , and initialize every unit with a random activation value [ 1:1 ] .
though values of v and e are chosen such that a network has a high probability of being connected , this is not required.update units .
synchronously update every unit 's activation value from its own and all its adjacent units ' activation values according to equation ( 1).select pivot and candidate .
randomly select one unit from the network ( the pivot ) . from all other units , select the one whose activation value is closest to the pivot 's .
then , from the units already adjacent to the pivot , select the one whose activation value is farthest from the pivot 's , and cut its connection to keep the number of connections constant .
if there is already a connection between the pivot and the candidate , or if the pivot has zero connections , nothing happens and this step is skipped.iteration completed .
randomly establish e connections between v units to create a ( v , e ) random network , and initialize every unit with a random activation value [ 1:1 ] .
though values of v and e are chosen such that a network has a high probability of being connected , this is not required .
synchronously update every unit 's activation value from its own and all its adjacent units ' activation values according to equation ( 1 ) .
randomly select one unit from the network ( the pivot ) . from all other units , select the one whose activation value is closest to the pivot 's .
then , from the units already adjacent to the pivot , select the one whose activation value is farthest from the pivot 's , and cut its connection to keep the number of connections constant .
if there is already a connection between the pivot and the candidate , or if the pivot has zero connections , nothing happens and this step is skipped .
networks implementing these iterative steps exhibit development from an initial random configuration to modular small - world configurations ( gong and van leeuwen , 2003 ; van den berg and van leeuwen , 2004 ; rubinov et al . , 2009b )
but as it turns out , both the consistent build - up of connective modularity on one hand , or the loss of structural coherence due to functional fragmentation on the other , are a result of changing dynamic activity depending critically on the number of connections in the network .
the influence of these numbers shows a close relationship to the percolation function of random graphs .
a common principle for neural network evolution is preferential attachment ( barabsi and albert , 1999 ) .
this mechanism leads to networks that are scale - free , but not modular small - worlds . only by combining preferential attachment with adaptive , hebbian rewiring ,
does a network emerge that is scale - free and also has modular small - world network structure ( gong and van leeuwen , 2003 ) . an adaptive rewiring scenario for evolving networks allows networks with initially random or regular structures to develop into modular small - world structures ( gong and van leeuwen , 2004 ; rubinov et al . , 2009b ) .
the scenario requires network units ( edges ) that produce ongoing , non - random , non - periodic oscillatory activity .
these could , for instance , be represented by spiking model neurons ( kwok et al . , 2007 ) or by nonlinear maps as an extremely simplified model of neural mass activity ( breakspear et al .
, 2003a , b ) . with these simple units as edges , the vertices of the network represents the couplings of a coupled nonlinear map ( kaneko , 1989 ) .
adaptive rewiring operates on this activity according to the general hebbian principle of  what fires together wires together  ( paulsen and sejnowski , 2000 ) . at successive points during the systems ongoing spontaneous activity
, connections are added between pairs of synchronously active but hitherto unconnected units , while connections between desynchronized units are removed ( see methods ) . over time
the network gradually assumes a modular , small - world structure ( figure 1 ) .
adaptive rewiring leads from an initial random network ( left ) , to modular small - world structure ( right ) in small iterative steps . coupled chaotic oscillators intermittently synchronize and desynchronize their activity spontaneously in patterns of great variability . after some time a momentarily synchronized pair of units that are not connected
receive a connection , which is removed from a pair that are connected but not synchronized . as this process continues , a modular , small - world structure emerges from an initially random configuration . to obtain a more detailed view of this phase transition , we use the adaptive rewiring scenario with coupled nonlinear maps ( kaneko , 1989 ) with initially randomly structured graphs , for a range of different numbers of vertices v : v = 300 , 400 , 500,  ,1000 vertices and numbers of edges e that differ by small steps of 20 .
for each combination of v , e , across four million iterations we measured the cc and the cpl every one thousand iterations , resulting in a 4000 point record for each of five runs .
the maximum , minimum , and mean values of the last 2000 points in each run were averaged over the five runs as illustrated in figure 3 .
meanwhile a mixture of regular and irregular behavior is established in the network activity that is itself optimal for sustaining the small - world structure .
crucially , whilst low dimensional , ordered , and synchronized activity dominates within modular communities , high dimensional unsynchronized activity in connector hubs ensures that the system does not fragment ( rubinov et al .
attractors  in the space of possible systems ( gong and van leeuwen , 2004 ) , which offers a potential explanation for their ubiquity in biological neural networks at different scales , including the entire brain ( barabsi and albert , 1999 ) . in this scenario , connectivity constitutes a critical limit for the evolution to small - world structure ( figure 2 ) .
when the number of edges is large enough , adaptive rewiring guarantees a robust evolution from random to small - world connectivity . below this limit , this evolution is frustrated , and fails to reach a stable asymptotic state . with reduced connectivity levels , we first encounter critical fluctuation : intermittently during some episodes , clusterings are formed intermittently , which are annihilated in other episodes
. this may reflect the intermittent occurrence of certain symptoms ( e.g. , delusions ) as the brain disease first becomes manifest .
for still lower connectivity levels , adaptive rewiring becomes completely ineffective ; this may reflect the advanced state of the disease .
self - organization from random to small - world critically in a network of 700 vertices .
the self - organization occurs through adaptive rewiring . whether a small - world emerges depends on the number of edges .
we compared the critical limit on the evolution to small - world structures to percolation thresholds of random networks with the same numbers of edges and vertices .
figure 4 shows that the observed minimum cc can be modeled as a linear function of cp(n ) , with k3 for offset and k4 for amplitude : ccpred = k3 + k4 
 cp(n ) .
parameter k3 was in the range [ 0.107:0.196 ] , parameter k4 in [ 0.392:0.459 ] and parameter k1 in [ 0.001:0.006 ] .
the behavior of these parameters across network sizes was not monotonic ( figure 4 ) .
parameter k2 however , the horizontal position of the anchor point , showed a universal scaling law to the anchor point in the percolation function of random graphs , namely ( table 1 ) : aswn(n ) = arand(n ) .
note : anchor point arand(n ) =   n  ln(n ) for classic random graphs of n vertices ; this anchor point indicates the percolation threshold , where the percolation function cp(n ) shows the greatest inflection .
aswn(n ) : anchor point for the small - world networks fitted according to figure 4 .
marquardt algorithm implemented in fityk . scaling power : the value of h in the equation aswn(n ) = arand(n ) .
a common principle for neural network evolution is preferential attachment ( barabsi and albert , 1999 ) .
this mechanism leads to networks that are scale - free , but not modular small - worlds . only by combining preferential attachment with adaptive , hebbian rewiring ,
does a network emerge that is scale - free and also has modular small - world network structure ( gong and van leeuwen , 2003 ) . an adaptive rewiring scenario for evolving networks allows networks with initially random or regular structures to develop into modular small - world structures ( gong and van leeuwen , 2004 ; rubinov et al . , 2009b ) .
the scenario requires network units ( edges ) that produce ongoing , non - random , non - periodic oscillatory activity .
these could , for instance , be represented by spiking model neurons ( kwok et al . , 2007 ) or by nonlinear maps as an extremely simplified model of neural mass activity ( breakspear et al .
, 2003a , b ) . with these simple units as edges , the vertices of the network represents the couplings of a coupled nonlinear map ( kaneko , 1989 ) .
adaptive rewiring operates on this activity according to the general hebbian principle of  what fires together wires together  ( paulsen and sejnowski , 2000 ) . at successive points during the systems ongoing spontaneous activity
, connections are added between pairs of synchronously active but hitherto unconnected units , while connections between desynchronized units are removed ( see methods ) . over time
the network gradually assumes a modular , small - world structure ( figure 1 ) .
adaptive rewiring leads from an initial random network ( left ) , to modular small - world structure ( right ) in small iterative steps . coupled chaotic oscillators intermittently synchronize and desynchronize their activity spontaneously in patterns of great variability . after some time a momentarily synchronized pair of units that are not connected
receive a connection , which is removed from a pair that are connected but not synchronized . as this process continues , a modular , small - world structure emerges from an initially random configuration . to obtain a more detailed view of this phase transition , we use the adaptive rewiring scenario with coupled nonlinear maps ( kaneko , 1989 ) with initially randomly structured graphs , for a range of different numbers of vertices v : v = 300 , 400 , 500,  ,1000 vertices and numbers of edges e that differ by small steps of 20 .
for each combination of v , e , across four million iterations we measured the cc and the cpl every one thousand iterations , resulting in a 4000 point record for each of five runs .
the maximum , minimum , and mean values of the last 2000 points in each run were averaged over the five runs as illustrated in figure 3 .
meanwhile a mixture of regular and irregular behavior is established in the network activity that is itself optimal for sustaining the small - world structure .
crucially , whilst low dimensional , ordered , and synchronized activity dominates within modular communities , high dimensional unsynchronized activity in connector hubs ensures that the system does not fragment ( rubinov et al .
attractors  in the space of possible systems ( gong and van leeuwen , 2004 ) , which offers a potential explanation for their ubiquity in biological neural networks at different scales , including the entire brain ( barabsi and albert , 1999 ) . in this scenario , connectivity constitutes a critical limit for the evolution to small - world structure ( figure 2 ) .
when the number of edges is large enough , adaptive rewiring guarantees a robust evolution from random to small - world connectivity . below this limit , this evolution is frustrated , and fails to reach a stable asymptotic state . with reduced connectivity levels , we first encounter critical fluctuation : intermittently during some episodes , clusterings are formed intermittently , which are annihilated in other episodes
. this may reflect the intermittent occurrence of certain symptoms ( e.g. , delusions ) as the brain disease first becomes manifest .
for still lower connectivity levels , adaptive rewiring becomes completely ineffective ; this may reflect the advanced state of the disease .
self - organization from random to small - world critically in a network of 700 vertices .
the self - organization occurs through adaptive rewiring . whether a small - world emerges depends on the number of edges .
we compared the critical limit on the evolution to small - world structures to percolation thresholds of random networks with the same numbers of edges and vertices .
figure 4 shows that the observed minimum cc can be modeled as a linear function of cp(n ) , with k3 for offset and k4 for amplitude : ccpred = k3 + k4 
 cp(n ) .
parameter k3 was in the range [ 0.107:0.196 ] , parameter k4 in [ 0.392:0.459 ] and parameter k1 in [ 0.001:0.006 ] .
the behavior of these parameters across network sizes was not monotonic ( figure 4 ) .
parameter k2 however , the horizontal position of the anchor point , showed a universal scaling law to the anchor point in the percolation function of random graphs , namely ( table 1 ) : aswn(n ) = arand(n ) .
note : anchor point arand(n ) =   n  ln(n ) for classic random graphs of n vertices ; this anchor point indicates the percolation threshold , where the percolation function cp(n ) shows the greatest inflection .
aswn(n ) : anchor point for the small - world networks fitted according to figure 4 .
marquardt algorithm implemented in fityk . scaling power : the value of h in the equation aswn(n ) = arand(n ) .
we propose that important insights into cortical activity and architecture can be obtained by modeling the activity - dependent rewiring of neural connections during development ( gong and van leeuwen , 2003 ; rubinov et al . , 2009b ) .
in our model , network connections evolve in accordance with the principle that the structure rewires in adaptation to spontaneous , on - going activity .
this evolution , however , is only guaranteed if there are sufficiently many connections available .
if connectivity is reduced below this number , the structure shifts toward randomness ; in particular , local clustering is reduced .
andreasen ( 1999 ) and friston and frith ( 1995 ) considered schizophrenia as fragmentation , understood as the breakdown of integration between widely distributed brain areas ( stephan et al . , 2006 , 2009 ) .
this breakdown can be associated with the loss of connectivity ( zalesky et al . , 2011 ) , in particular of input to layer 3 pyramidal cells , an effect which is well - documented ( e.g. , garey et al .
( 2011 ) observed widespread impairment in structural connectivity in schizophrenic patients , involving medial frontal , parietal / occipital and left frontal cortex .
it should be observed that the loss of connectivity that may lead to the onset of schizophrenia can be relatively subtle . across the population ,
inputs to layer 3 pyramidal cells are substantially reduced during late adolescence , the typical period for the onset of schizophrenia ( bourgeois et al . , 1994 ) .
given that brain connectivity is costly , it may well be that in normals , its density hovers just above the critical level ( the anchor point in figure 4 ) , but in early schizophrenia it may fall just below this point . the graph - theoretical concept of percolation tells us that a small decline in connectivity can lead to a sudden breakdown of global network coherence .
based on our results , however , we argue that fragmentation in brain pathologies such as schizophrenia may be considered theoretically as a breakdown in the local connectivity structure , prior to the loss of global coherence . the number minimally needed to secure local modularity , and hence to prevent it from shifting toward randomness in structure , is systematically related to , and greater than , that needed to secure global connectivity , even if the system has fallen into entirely random connectivity .
this result is of potential importance for understanding the pathophysiological processes that give rise to this disorder .
the loss of local clustering in our model is in accordance with observations in schizophrenic patients by micheloyannis et al .
et al . ( 2006 ) , the clinical group also showed longer path lengths than the controls , whereas in rubinov et al .
nevertheless , we might attribute the discrepancy to the fact that in both studies comparisons were made , for statistical reasons , between networks that were thresholded to have identical connectivity .
whereas the above - mentioned effects of clustering remain relatively unaffected by threshold setting , the differences in path length rapidly disappear for lower thresholds ( figure 1 in micheloyannis et al . , 2006 ) .
( 2003b ) reported that although there were no significant increases in the occurrence of nonlinear interdependence between pairs of electrodes in schizophrenia , there was an increase in the co - occurrence in multiple ( widespread ) instances of nonlinear interdependence .
this means that a relatively large number of global connections will have survived thresholding in rubinov 's study , leading to their observation of path length shortening .
it can not be concluded from rubinov 's study , therefore , that global connectivity is stronger in schizophrenics than in normals ; it could , however , be concluded that the global connectivity becomes stronger in schizophrenia relatively to their local connectivity .
such a conclusion would entirely be in accordance with the modularity breakdown observed in our model .
along the lines set out here , a shift in the balance from local to global connectivity is perfectly consistent with an overall loss of connectivity in early schizophrenia .
( 2003 ) introduced the notion of  overbindingthe formation of excessive connections that are effectively random  and , as such , do not enable distinguishing external from internal sources , thus providing conditions favorable for phenomena such as hallucination . a possible objection to our findings is the specific choice of our rewiring algorithm .
note , however , that in the present paper we sought to establish the principled possibility using the simplest possible model , rather than to establish the empirical validity through the most realistic model possible
. note that , as a consequence , the model contains only generic dynamical and adaptive principles .
we have discussed elsewhere the robustness of this model ( gong and van leeuwen , 2003 , 2004 ; van den berg and van leeuwen , 2004 ; kwok et al .
an important limitation is that the model inevitably makes over - simplifying assumptions . in particular
inter - modular connections are physically of longer range than intra - modular ones and , therefore , have a higher metabolic cost and a greater vulnerability .
preliminary analysis of models with more realistic constraints does not appear , however , to affect our conclusions .
clearly , a more differentiated model is needed to address empirical datasets such as ( rubinov et al . , 2009a ) , an important goal of future work .
however , it should also be noted that uncovering universal principles  such as those reported here  has the advantage of being 
detail invariantthat is , robust across a range of potential constraints , whereas findings arising in detailed models may not be robust to changes in those details .
we observed universal scaling behavior in adaptive self - organization of clustered small - world networks : the connectivity needed for these network properties to emerge under hebbian rewiring scales with a universal power  = 1.17 to the percolation function in random networks .
note , first , that  > 1 might have been expected , given that the requirement to observe clustering and small - world structure are constraints additional to percolation .
what is surprising is that these requirements are met with alpha very close to unity ; near - linear scaling implies that these additional constraints can be realized with great efficiency . in terms of kolmogorov - complexity , small - worlds are compressible , whereas almost every possible network of n nodes and e edges ( or equivalently a bit string of length l = n(n  1 ) with e ones and l - e zeros ) will be incompressible ( li and vitnyi , 1993 ) . in this perspective , the ubiquity of small - world structure in real - world networks is quite astonishing : within human brains ( sporns and zwi , 2004 ; stam , 2004 ; eguluz et al .
bartolomei et al . , 2006 ; micheloyannis et al . , 2006 ; ponten et al .
, 2007 ; rubinov et al . , 2009a ; bassett et al . , 2010 ) , as well as between them : networks of scientific co - authorship ( newman , 2001 ) , collaborating movie actors ( watts and strogatz , 1998 ; amaral et al . , 2000 ) , social networks in general ( wasserman and faust , 1994 ) . here
we showed how such a network could arise with minimal connectivity close to random network percolation what is the reason for the universality of the scaling exponent ?
we may wonder whether the same exponent found in other domains , could help us understand the principle .
a study of class graphs in open - source , object - oriented software systems ranging from simple paint programs , peer - to - peer downloaders , racing games , database management software to a complete operating system , showed that the number of links between classes scales to the number of classes with an exact power = 1.17 .
the authors found that class graphs are small - world networks at the critical threshold for the breakdown of modularity , which happens when developments to the system are widely dispersed and affect many unrelated classes in apparently distant modules ( valverde and sol , 2007 ) .
the similarity of this finding to ours supports the view that the scaling exponent reflects a general feature in the emergence and breakdown of modular network structure .
the study of self - organizing modular small - world networks casts a new perspective on psychiatric illnesses characterized by disorganized cognition , such as schizophrenia , of which the expression has been attributed to fragmentation  a  subtle but pernicious disconnection  ( friston , 1996 , p. 644 ) . rather than a breakdown in global connectivity
, we propose that fragmentation is to be understood as a failure to organize the functional connectivity of the brain into a modular small - world structure .
this is in accordance with the observed  random shift  in schizophrenic ( micheloyannis et al . , 2006 ;
random networks are considered extremely uneconomical ; in terms of cable length , an optimal configuration combines local modules with a limited number of large - scale connections ( murre and sturdy , 1995 ) . even though our model does not consider distance , in terms of network topology it is still the case that information travels efficiently both within locally connected circuits of modular small - world graphs and between their circuits , which makes these networks efficient for transport or communication ( latora and marchiori , 2001 ; bassett et al . , 2010 ) .
the scaling observations tell us that fragmentation is a result of a breakdown in local , rather than global structure . with progressive loss of connectivity ,
ultimately , it may not matter which connections are lost first , the result may be a cascade of changes that lead to the network falling apart . for diagnosis , however , a proper understanding of the early stages of the disease is crucial ; loss of modularity might offer a new perspective on the origins of the disease .
one simulation consists of one network of v units and e connections , which is randomly initialized and then iterated exactly 4,000,000 times , simultaneously rearranging its connections and activity patterns , according to the adaptive rewiring scenario .
the smallest simulation we adopt has v = 300 units and e = 2400 connections . during iteration ,
its cpl and its cc are taken every 1000th iteration ( 1000 , 2000 ,  , 40,00,000 ) resulting in a 4000 point record , with a value for cc and a value for cpl at each point .
although the speed of convergence depends on the size of a network , 4 , 000 , 000 iterations prove to be enough to clearly discern asymptotic behavior for all simulations used in this investigation ( figure 3 ) .
evolution under adaptive rewiring of maximum , minimum , and average cluster coefficient and characteristic path length .
( a ) the values of minimum , maximum and average cc for networks of 700 vertices and edges ranging from [ 7000 , 7020 , 7040,  ,10,000 ] after extensive adaptive rewiring . note that beyond 9000 edges , cc - values tighten to a narrow range , indicating strong and consistent clustering behavior .
( b ) the values of minimum , maximum , and average cpl for networks of 700 vertices and edges ranging from [ 7000 , 7020 , 7040,  ,10,000 ] after extensive adaptive rewiring . beyond 9000 edges
thus , for 700 vertices , at least 9000 edges are needed for adaptive rewiring to converge to small - world structure . from the 4000 point record ,
the maximum , minimum , and average values for both cpl and cc are calculated from the last 2000 points . for statistical robustness
, we do any single simulation five times , and average the five values over this simulation - quintuple , resulting in a maximum , minimum , and average cc and cpl for the ( v = 300 , e = 2400 ) network .
we then start a new quintuple of simulations , increasing the number of connections e by 20 , generating five networks with v = 300 units and e = 2420 connections , and calculate the maximum , minimum , and average cc and cpl values from these five new simulations .
we keep starting new quintuples , repeatedly increasing e by 20 , until e = 3300 and the batch of 300-quintuples is complete . from the entire batch , the six cc and cpl values of every ( v ,
e ) are taken to graph the asymptotic clustering and path - length behavior of networks of 300 units as it depends on the numbers of connections ( figure 4 ) .
gray lines represent minimal , maximal and average observed values for clustering coefficient , the dotted line is the predicted clustering coefficient , ccpred , a linear function of the percolation function cp(n ) of a random graph of n vertices : ccpred = k3 + k4  cp(n ) fitted with parameters k3 and k4 to the minimum observed clustering ; the arrow indicates its anchor point aswn(n ) with the corresponding number of edges in parentheses .
this process is then repeated for a batch of quintuples of networks with 400 units and numbers of connections 3600 , 3620 ,  , 5000 ( see figure 4 , top - right box ) .
we continue doing this for batches of networks with 500 , 600 , 700 , 800 , 900 , and 1000 units , with connections increasing by 20 , showing asymptotic clustering and path - length behavior depending on connectivity for networks of different sizes ( figure 4 ) .
note that although for 300 units , connections ranged from e = 2400 to e = 3300 , these numbers are different for larger networks .
for each of the eight batches , a phase transition was witnessed for both the cc and the cpl .
to pin down the exact location of the steepest inclination in the phase transition of the cc ( its center , or  anchor point  ) , the percolation function for classic random graphs was function - matched to the minimum cc of every batch .
the minimum cluster coefficient was chosen over the average and the maximum cluster coefficient because it has the steepest inclination , which facilitates the fitting best .
macquardt an iterative curve - fitting algorithm which operates by minimizing the summed squares of the residuals , in this case the difference between minimal cc - values of n - edge simulation quintuples on the one hand , and the ( erds and rnyi , 1959 ) percolation function 's value for n edges on the other .
macquardt algorithm is sensitive to local minima which makes it inefficient when using completely random initial values .
initial parameters were hand - guessed separately for each of the eight subgraphs in figure 4 , after which the algorithm was ran until convergence beyond the program 's six - digit resolution , a procedure that was repeated three times with small differences in the hand - guessed initial parameters .
the final values did not differ within the program 's six - digit resolution over the three repetitions , and convergence was very fast ( typically well before 100 iterations ) .
the fits show significant deviations from the data curve , due to intrinsic fluctuations in the data .
we , therefore , considered reliable the estimates of the scaling power and other model parameters .
even more reliable estimates could , in principle , be obtained by scaling up the network size to 2000 , 5000 , and 10,000 vertices , resources permitting , as computation time and data grow nonlinearly with network size .
the authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest .
the authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest .
 psychiatric illnesses characterized by disorganized cognition , such as schizophrenia , have been described in terms of fragmentation and hence understood as reduction in functional brain connectivity , particularly in prefrontal and parietal areas . however , as graph theory shows , relatively small numbers of nonlocal connections are sufficient to ensure global coherence in the modular small - world network structure of the brain . 
 we reconsider fragmentation in this perspective . 
 computational studies have shown that for a given level of connectivity in a model of coupled nonlinear oscillators , modular small - world networks evolve from an initially random organization . here 
 we demonstrate that with decreasing connectivity , the probability of evolving into a modular small - world network breaks down at a critical point , which scales to the percolation function of random networks with a universal exponent of  = 1.17 . 
 thus , according to the model , local modularity systematically breaks down before there is loss of global coherence in network connectivity . 
 we , therefore , propose that fragmentation may involve , at least in its initial stages , the inability of a dynamically evolving network to sustain a modular small - world structure . 
 the result is in a shift in the balance in schizophrenia from local to global functional connectivity . 
Introduction
Methods
Results
Evolving networks
Percolation and self-organization in small-worlds
Discussion
Data
Conflict of interest statement