given data , for , with a -dimensional real - valued predictor variable , where , and a real - valued response , a typical goal of regression analysis is to find an estimator , such that the expected loss is minimal , under a given loss function .for the following , the standard squared error loss is used .if the predictor can be of the ` black - box ' type , tree ensembles have proven to be very powerful .random forests is a prime example , as are boosted regression trees .there are many interesting tools available for interpretation of these tree ensembles , see for example and the references therein .while tree ensembles often have very good predictive performance , an advantage of a linear model is better interpretability .measuring variable importance and performing variable selection are more easier to formulate and understand in the context of linear models . for high - dimensional data with ,regularization is clearly imperative and the lasso has proven to be very popular in recent years , since it combines a convex optimization problem with variable selection .a precursor to the lasso was the nonnegative garrote .a disadvantage of the nonnegative garrote is the reliance on an initial estimator , which could be the least squares estimator or a regularized variation . on the positive side ,important variables incur less penalty and bias under the regularization than they do with the lasso . for a deeper discussion of the properties of the nonnegative garrote see .  here, it is proposed to use random forest as an initial estimator for the nonnegative garrote .the idea is related to the rule ensemble approach of , who used the lasso instead of the nonnegative garrote .a crucial distinction is that rules fulfilling the same functional role are grouped in our approach .this is similar in spirit to the group lasso .this produces a very accurate predictor that uses just a few functional groups of rules , discarding many variables in the process as irrelevant .a unique feature of the proposed method is that is seems to work very well in the absence of a tuning parameter .it just requires the choice of an initial tree ensemble .this makes the procedure very simple to implement and computationally efficient .the idea and the algorithm is developed in section [ section:2 ] , while a detailed numerical study on 15 datasets makes up section [ section:3 ] .a tree is seen here as a piecewise - constant function derived from the tree structure in the sense of . proposed ` rules ' as a name for simple rectangular - shaped indicator functions .every node in a tree is associated with a in -dimensional space , defined as the set of all values that pass through node if passed down the tree .all values that do not pass throuh node are outside of .the way rules are used here , they correspond to indicator functions , i.e.  is the indicator function for box . for a more detailed discussionsee .  to give an example of a rule , take the well - know dataset on abalone as an example .the goal is to predict age of abalone from physical measurements . for each of the 4177 abalone in the dataset , eight predictor variables ( sex , length , diameter , height , while weight , shucked weight , viscera weight and shell weight ) are available .an example of a rule is and the presence of such a rule in a final predictor is easy to interpret , comparable to interpreting coefficients in a linear model . for the following, it is assumed that rules contain at most a single inequality for each variable . in other words ,the boxes defined by rules in -dimensional spaces are defined by at most a single hyperplane in each variable .if a rule violates this assumption , it can easily be decomposed into two or several rules satisfying the assumption .every regression tree can be written as a linear superposition of rules .suppose a tree has nodes in total .the regression function of this tree ( ensemble ) can then be written as for some .the decomposition is not unique in general .we could , for example , assign non - zero regression coefficients only to leaf nodes . here, we build the regression function incrementally instead , assigning non - zero regression coefficients to _ all _ nodes .the value are defined as where is the empirical mean across the observations and is the parent node of in the tree .rule ( [ ruleex ] ) , in the abalone example above , receives a regression weight .the contribution of rule ( [ ruleex ] ) to the random forest fit is thus to increase the fitted value if and only if the diameter is larger than 0.537 and shell weight is larger than 0.135 .the random forest fit is the sum of the contribution from all these rules , where each rule corresponds to one node in the tree ensemble .  to see that ( [ treehat ] ) and ( [ eq : hatbetatree ]) really correspond to the original tree ( ensemble ) solution , consider just a single tree for the moment .denote the predicted value for predictor variable by .let be the rectangular area in -dimensional space that corresponds to the leaf node of the tree in which predictor variable falls .the predicted value follows then by adding up all relevant nodes and obtaining with ( [ treehat ] ) and ( [ eq : hatbetatree ] ) , i.e. the predicted values is just the empirical mean of across all observations which fall into the same leaf node as , which is equivalent to the original prediction of the tree .the equivalence for tree ensembles follows by averaging across all individual trees as in ( [ treehat ] ) .the idea of is to modify the coefficients in ( [ treehat ] ) , increasing sparsity of the fit by setting many regression coefficient to 0 and eliminating the corresponding rules from the fit , while hopefully not degrading the predictive performance of the tree ensemble in the process .the rule ensemble predictors are thus of the form sparsity is enforced by penalizing the -norm of in lasso - style , this enforces sparsity in terms of rules , i.e. the final predictor will have typically only very few rules , at least compared to the original tree ensemble .the penalty parameter is typically chosen by cross - validation . to add the linear main effects of all variables into ( [ ruleens ] ) , which were omitted here for notational simplicity and to keep invariance with respect to monotone transformations of predictor variables .it is shown in that the rule ensemble estimator maintains in general the predictive ability of random forests , while lending itself more easily to interpretation .    _first column : combined variable interaction rules between predictor variable _ tch _ ( x - axis to the right ) and _ ltg _ ( y - axis to the left ) in the random forest fit for the diabetes data .second column : the interaction rules can be decomposed into the effects ( plotted on z - axis ) of four interaction patterns on top , on second from top , on second from bottom and on the bottom .third column : applying the garrote correction , three of the four interaction patterns are set to 0 .fourth column : adding the four interaction patterns with garrote correction up , the total interaction pattern between the two variables _ ltg _ and _ tch _ in the forest garrote fit . _ _ , title="fig:",scaledwidth=25.0% ]  _ first column : combined variable interaction rules between predictor variable _ tch _ ( x - axis to the right ) and _ ltg _ ( y - axis to the left ) in the random forest fit for the diabetes data .second column : the interaction rules can be decomposed into the effects ( plotted on z - axis ) of four interaction patterns on top , on second from top , on second from bottom and on the bottom .third column : applying the garrote correction , three of the four interaction patterns are set to 0 .fourth column : adding the four interaction patterns with garrote correction up , the total interaction pattern between the two variables _ ltg _ and _ tch _ in the forest garrote fit . _ _ , title="fig:",scaledwidth=24.0% ]  _ first column : combined variable interaction rules between predictor variable _ tch _ ( x - axis to the right ) and _ ltg _ ( y - axis to the left ) in the random forest fit for the diabetes data .second column : the interaction rules can be decomposed into the effects ( plotted on z - axis ) of four interaction patterns on top , on second from top , on second from bottom and on the bottom . third column : applying the garrote correction , three of the four interaction patterns are set to 0 .fourth column : adding the four interaction patterns with garrote correction up , the total interaction pattern between the two variables _ ltg _ and _ tch _ in the forest garrote fit ._ _ , title="fig:",scaledwidth=24.0% ]  _ first column : combined variable interaction rules between predictor variable _ tch _ ( x - axis to the right ) and _ ltg _ ( y - axis to the left ) in the random forest fit for the diabetes data .second column : the interaction rules can be decomposed into the effects ( plotted on z - axis ) of four interaction patterns on top , on second from top , on second from bottom and on the bottom . third column : applying the garrote correction , three of the four interaction patterns are set to 0 .fourth column : adding the four interaction patterns with garrote correction up , the total interaction pattern between the two variables _ ltg _ and _ tch _ in the forest garrote fit . _ _ , title="fig:",scaledwidth=25.0% ]  the rule ensemble approach is treating all rules equally by enforcing a -penalty on all rules extracted from a tree ensemble .it does not take into account , however , that there are typically many very closely related rules in the fit .take the rf fit to the abalone data as an example .several hundred rules are extracted from the rf , two of which are with regression coefficient and with regression coefficient .the effect of these two rules , measured by and , are clearly very similar .in total , there are 32 rules ` interaction ' rules that involve variables diameter and shell weight in the rf fit to the abalone data .selecting some members of this group , it seems artificial to exclude others of the same ` functional ' type .sparsity is measured purely on a rule - by - rule basis in the -penalty term of rule ensembles ( [ ruleens ] ) .selecting the two rules mentioned above incurs the same sparsity penalty as if the second rule involved two completely different variables .an undesirable side - effect of not taking into account the grouping of rules is that many or even all original predictor variables might still be involved in the rules ; sparsity is not explicitly enforced in the sense that many irrelevant original predictor variables are completely discarded in the selected rules .it seems natural to let rules form functional groups .the question then turns up which rules form useful and interpretable groups .there is clearly no simple right or wrong answer to this question . here, a very simple yet hopefully intuitive functional grouping of rules is employed . for the -th rule , with coefficient ,define the interaction pattern for variables by    the meaning of interaction patterns is best understood if looking at examples for varying degrees , where the degree of a interaction pattern is understood to be the number of non - zero entries in and corresponds to the number of variables that are involved in a rule .[ [ first - degree - main - effects . ] ] first degree ( main effects ) .+ + + + + + + + + + + + + + + + + + + + + + + + + + + +  the simplest interaction patterns are those involving a single variable only , which correspond in some sense to the main effects of variables . the interaction pattern for example collects all rules that involve the 2nd predictor variable _ length _ only and lead to a monotonically increasing fit ( are thus of the form for some real - valued if the corresponding regression coefficient were positive or if the regression coefficient were negative ) . the interaction pattern collects conversely all those rules that yield a monotonically decreasing fit in the variable _ diameter _ , the third variable .[ [ second - degree - interactions - effects . ] ] second degree ( interactions effects ) .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  second degree interaction patterns are of the form ( [ ruleex1 ] ) or ( [ ruleex2 ] ) .as diameter is the 3rd variable and shell weight the 8th , the interaction pattern of both rules ( [ ruleex1 ] ) and ( [ ruleex2 ] ) is making them members of the same functional group , as for both rules the fitted value is monotonically increasing in both involved variables .in other words , either a large value in both variables increases the fitted value or a very low value in both variables decreases the fitted value .second degree interaction patterns thus form four categories for each pair of variables .a case could be made to merge these four categories into just two categories , as the interaction patterns do not conform nicely with the more standard multiplicative form of interactions in linear models .however , there is no reason to believe that nature always adheres to the multiplicative form of interactions typically assumed in linear models .the interaction patterns used here seemed more adapted to the context of rule - based inference .factorial variables can be dealt with in the same framework ( [ sigma ] ) by converting to dummy variables first .in contrast to the group lasso approach of , the proposed method does not only start with knowledge of natural groups of variables or rules .a very good initial estimator is available , namely the random forest fit .this is exploited in the following .let be the part of the fit that collects all contributions from rules with interaction pattern , let be the collection of all possible interaction patterns .the tree ensemble fit ( [ treehat ] ) can then be re - written as a sum over all interaction patterns a interaction pattern is called _ active _ if the corresponding fit in the tree ensemble is non - zero , i.e. if and only if is not identically .the random forest fit contains very often a huge number of active interaction pattern , involving interactions up to fourth and higher degrees .most of those active patterns contribute just in a negligible way to the overall fit .the idea proposed here is to use ( [ tdecomp ] ) as a starting point and modify it to enforce sparsity in the final fit , getting rid of as many unnecessary predictor variables and associated interaction patterns as possible .the lasso of was used in the rule ensemble approach of . here , however , the starting point is the functional decomposition ( [ tdecomp ] ) , which is already a very good initial ( yet not sparse ) estimator of the underlying regression function .hence it seems more appropriate to use breiman s nonnegative garrote , penalizing contributions of interaction patterns less if their contribution to the initial estimator is large and vice versa .the beneficial effect of this bias reduction for important variables has been noted in , amongst others , and .the garrote - style forest estimator is defined as each contribution of an interaction pattern is multiplied by a factor .the original tree ensemble fit is obtained by setting all factors equal to 1 .the multiplicative factor is chosen by least squares , subject to the constraint that the total -norm of the multiplying coefficients is less than 1 , the normalizing factor divides the -norm of by the total number of interaction patterns and is certainly not crucial here but simplifies notation .the estimation of is an application of breiman s nonnegative garrote . as for the garrote ,the original predictors are not rescaled , thus putting effectively more penalty on unimportant predictors , with little variance of across the samples and less penalty on the important predictors with higher variance , see for details .algorithmically , the problem can be solved with an efficient lars algorithm , which can easily be adapted to include the positivity constraint .alternatively , quadratic programming can be used .it might be surprising to see the -norm constrained by 1 instead of a tuning parameter  .yet this is indeed one of the interesting properties of forest garrote .the tree ensemble is in some sense selecting a good level of sparsity .it seems maybe implausible that this would work in practice , but some intuitive reasons for its empirical success are given further below and ample empirical evidence is provided in the section with numerical results .a drawback of the garrote in the linear model setting is the reliance on the ols estimator ( or another suitable estimator ) , see also .the ols estimator is for example not available if .the tree ensemble estimates are , in contrast , very reasonable estimators in a wide variety of settings , certainly including the high - dimensional setting .the entire forest garrote algorithm works thus as follows  1 .fit random forest or another tree ensemble approach to the data .2 .  extract from the tree ensemble for all by first extracting all rules and corresponding regression coefficients and grouping them via ( [ tsigma ] ) for each interaction pattern .estimate as in ( [ gammahat ] ) from the data , using for example the lars algorithm .4 .  the fitted forest garrote function is given by ( [ hatgar ] ) .the whole algorithm is very simple and fast , as there is no tuning parameter to choose .   in most regularization problems , like the lasso , choosing the regularization parameter is very important and it is usually a priori not clear what a good choice will be , unless the noise level is known with good accuracy ( and it usually it is not ) . the most obvious approach would be cross - validation. cross - validation can be computationally expensive and is usually not guaranteed to lead to optimal sparsity of the solution , selecting many more variables or interaction patterns than necessary , as shown for the lasso in and .  since the starting point is a very useful predictor , the original tree ensemble, there is a natural tuning parameter for the forest garrote estimator .as noted before , corresponds to the original tree ensemble solution .the original tree ensemble solution is thus contained in the feasible region of the optimization problem ( [ gammahat ] ) under a constraint on the -norm of exactly 1 .the solution ( [ hatgar ] ) will thus be at least as sparse as the original tree ensemble solution in the sense if sparsity is measured in the same -norm sense as in ( [ gammahat ] ) .some variables might not be used at all even though they appear in the original tree ensemble .  on the other hand ,the empirical squared error loss of is at least as low ( and typically lower ) than for the original tree ensemble solution as will reduce the empirical loss among all solutions in the feasible region of ( [ gammahat ] ) , which contains the original tree ensemble solution .the latter point does clearly not guarantee better generalization on yet unseen data , but a constraint of 1 on the -norm turns out to be an interesting starting point and is a very good default choice of the penalty parameter .  sometimes one might still be interested in introducing a tuning parameter .one could replace the constraint of 1 on the -norm of in ( [ gammahat ] ) by a constraint , the range over which to search , with cross - validation , to over can then typically be limited to $ ] . empirically , it turned out that the default choice is very reasonable and actually achieves often better predictive and selection performance than the cross - validated solution , since the latter suffers from possibly high variance for finite sample sizes .   _ upper right diagonal : ` main effects ' and ` interactions ' of second degree for the random forest fit on the diabetes data between the main 6 variables ( not showing all variables ) .lower left diagonal : corresponding functions for the forest garrote .some main effects and interactions are set exactly to zero .vanishing interactions are not plotted , leaving some entries blank . _ _ , scaledwidth=99.0% ]   the method is illustrated on the diabetes data from with predictor variables , age , sex , body mass index , average blood pressure and six blood serum measurements .these variables were obtained for each of diabetes patients , along with the response of interest , a ` quantitative measure of disease progression one year after baseline ' . applying a random forest fit to this dataset, the main effects and second - order interactions effects , extracted as in ( [ tsigma ] ) , are shown in the upper right diagonal of figure [ fig : diabetes ] , for 6 out of the 10 variables ( chosen at random to facilitate presentation ) .all of these variables have non - vanishing main effects ( on the diagonal ) and the interaction patterns can be quite complex , making them somewhat difficult to interpret .  now applying a forest garrote selection to the random forest fit, one obtains the main effects and interaction plots shown in the lower left diagonal of figure [ fig : diabetes ] .note that the x - axis in the interaction plots corresponds to the variable in the same column , while the y - axis refers to the variable in the same row .interaction plots are thus rotated by 90 degrees between the upper right and the lower left diagonal . some main effects and interactionsare set to 0 by the forest garrote selection .interaction effects that are not set to 0 are typically ` simplified ' considerably .the same effect was observed and explained in figure [ fig:38 ] .the interaction plot of forest garrote seems thus much more amenable for interpretation .the unexplained variance on test data , as a fraction of the total variance . left : comparison of unexplained variance for forest garrote versus random forests .middle : forest garrote ( cv ) versus random forests .right : rule ensembles versus random forests ._ , scaledwidth=99.0% ]  ._ [ table : timetable ] the relative cpu time spent on forest garrote , forest garrote ( cv ) and rule ensembles for the various datasets .forest garrote uses the least computational resources since ( i ) it starts from a relative small set of dictionary elements ( all for as opposed to all rules ) , ( ii ) the solution has to be computed only for a single regularization parameter and there is hence ( iii ) also no need for expensive cross - validation .note that the times above are only for the rule selection steps ( [ ruleens ] ) and ( [ gammahat ] ) respectively and the overall relative speed difference is typically smaller as a tree ensemble fit needs to be computed as an initial estimator in all settings ._ [ cols="<,<,<,>,>,>",options="header " , ]   forest garrote ( cv ) has also obviously a computational disadvantage compared with the recommended forest garrote estimator , as shown in table [ table : timetable ] which is comparing relative cpu times necessary to compute the relative estimators .all three methods could be speeded up considerably by clever computational implementation .any such improvement would most likely be applicable to any of these three compared methods as they have very similar optimization problems at their heart .only relative performance measurements seem to be appropriate and only the time it takes to solve the respective optimization problems ( [ ruleens ] ) , ( [ gammahat ] ) and ( [ gammahatlambda ] ) is reported , including time necessary for cross - validation , if so required .rule ensembles is faring by far the worst here , since the underlying optimization problem is very high - dimensional .the dimensionality in ( [ ruleens ] ) is the total number of rules , which corresponds to the number of all nodes in the random forest fit .the total number of interaction patterns in the optimization underlying ( [ gammahat ] ) in the forest garrote fit is , on the other hand , very much smaller than the number of all rules , since many rules are typically combined in each interaction patterns .the lack of cross - validation for the forest garrote estimator clearly also speeds computation up by an additional factor between 5 and 10 , depending on which form of cross - validation is employed .finally , the number of variables selected by either method is examined .a variable is said to be selected for this purpose if it appears in any node in the forest or in any rule that is selected with a non - zero coefficient .in other words , selected variables will be needed to compute predictions , not selected variables can be discarded .the results are shown in table [ table : var ] .many variables are typically involved in a random forest fit and both rule ensembles as well as forest garrote can cut down this number substantially . especially for higher - dimensional data with large number of variables , the effect can be pronounced . between rule ensembles and forest garrote ,the differences are very minor with a slight tendency of forest garrote to produce sparser results .[ section : numerical ]balancing interpretability and predictive power for regression problems is a difficult act .linear models lend themselves more easily to interpretation but suffer often in terms of predictive power .random forests ( rf ) , on the other hand , are known deliver very accurate prediction .tools exist to extract marginal variable importance measures from rf .however , the interpretability of rf could be improved if the very large number of nodes in the hundreds of trees fitted for rf could be reduced .  here, the forest garrote was proposed as such a pruning method for rf or tree ensembles in general .it collects all rules or nodes in the forest that belong to the same functional group . using a garrote - style penalty ,some of these functional groups are then shrunken to zero , while the signal of other functional groups is enhanced .this leads to a sparser model and rather interpretable interaction plots between variables .predictive power is similar or better to the original rf fit for all examined datasets .the unique feature of forest garrote is that it seems to work very well without the use of a tuning parameter , as shown on multiple well known and less well known datasets .the lack of a tuning parameter makes the method very easy to implement and computationally efficient .

Abstract: variable selection for high - dimensional linear models has received a lot of attention lately , mostly in the context of -regularization . part of the attraction is the variable selection effect : parsimonious models are obtained , which are very suitable for interpretation . in terms of predictive power , however , these regularized linear models are often slightly inferior to machine learning procedures like tree ensembles . tree ensembles , on the other hand , lack usually a formal way of variable selection and are difficult to visualize . a garrote - style convex penalty for trees ensembles , in particular random forests , is proposed . the penalty selects functional groups of nodes in the trees . these could be as simple as monotone functions of individual predictor variables . this yields a parsimonious function fit , which lends itself easily to visualization and interpretation . the predictive power is maintained at least at the same level as the original tree ensemble . a key feature of the method is that , once a tree ensemble is fitted , no further tuning parameter needs to be selected . the empirical performance is demonstrated on a wide array of datasets .


the earliest accounts of mathematical modelling to capture the spread of diseases dates back to as early as the century .bernoulli used mathematical equations to defend his stand on vaccination against the outbreak of smallpox .works following bernoulli earliest formulation of epidemic modelling helped in understanding germ theory in detail .however , this was not until the works of mckendrick and kermack , which first proposed a deterministic model that predicted epidemic outbreaks very similar to the ones that were recorded during those times  .since then our understanding of the mathematical models in epidemology has evolved over the years , the accounts of which can be found in the extensive works of anderson and may  .all the above formulations focused on modelling epidemics over a set of population in which uniform ties between agents were assumed _ a priori_. also , the nature of the ties in the above models did not play any significant role .contrary to this , the enormous body of work during the last decade in the field of network science has asserted the fact that ties , their strength and types in a system or a population play a significant role  .thus , a system or a population is not considered to be a set of individuals , rather it is considered to be a complex network of interacting individuals , where each individual or an entity is considered to be a node , and the links between them define the type of relationship one shares with the other .interestingly , a network model is not exhaustive to the study of population , rather it is a universal framework which can be used to understand numerous complex systems in general .also , over the years the term _ epidemic modelling _ has evolved into a common metaphor for a wide array of dynamical processes on these networks .various complex phenomena such as percolation , the spreading of blackouts to the spreading of memes , ideas and opinions in a social network can be modelled under the common framework of epidemic modelling  .transportation networks play a vital role in the spread of epidemics due to their widespread outreach across cities , countries and continents  .hould people be worried about getting ebola on the subway was one of the numerous similar headlines that made the front pages of the newspapers around the world during the 2014 ebola scare  . in this particular incident however , nobody was infected because the subject did not show symptoms of ebola while using public transportation .therefore not only airline networks , that can transmit pathogens across continents , even modes of public transport operating within cities such as , buses and subways pose a serious threat as well as a source of panic during desperate times . although , epidemic spreading in airline networks have been studied extensively , similar studies on bus networks are relatively rare  .epidemological models have been simulated on bus network datasets ; however , the results were only used to validate the numerical models .also , a recent study on city - wide integrated travel networks ( itn ) , have found both the traveling speed and frequency to be important factors in epidemic spreading  .thus , the effect of network structure and constraints in epidemic spreading are yet to be studied in these networks . in this paperwe exploit the structural aspect of these networks to understand their role in epidemic spreading and diffusion process in detail .in this paper , we simulate the epidemic model on the bus networks of six major indian cities , namely ahmedabad ( abn ) , chennai ( cbn ) , delhi ( dbn ) , hyderabad ( hbn ) , kolkata ( kbn ) and mumbai ( mbn )  .the route data for all these bus networks are obtained from their respective state government websites .a bus network in -space can be considered a graph , , where denotes the set of nodes and , the set of links .the topological structure of the graph is generated by considering every stop as a node , and the routes connecting the stops form the set of links .thus , a bus network can be represented as a adjacency matrix , whose elements , if and are connected , else .the structure of a network in general is dependent upon the way the nodes are connected in a network .however , bus networks fall into a special class of complex networks where the physical constraints offered by the roads and the cities result in the emergence of the network topology  . the statistical analysis of these networks reveal a wide spectrum of topological structure .we define the degree , of a node , as .the pattern of the inter - nodal connectivity is given by the degree - distribution function , , which can be defined as the probability of a node , having a degree of atleast .some of the other network metrics that are of particular interest are the clustering coefficient , which denotes the tendency of nodes to form clusters or cliques and the characteristic path length .the local clustering coefficient is given by , where is the link connecting node pair , and are the neighbours of the node .the neighbourhood for a node , is defined as the set of its immediately connected neighbours , as , and the average clustering coefficient for the entire network is given by .the characteristic path length , is defined as the average number of nodes crossed along the shortest paths for all possible pairs of network nodes . the average distance from a certain vertex to every other vertex is given by . then , is calculated by taking the median of all the calculated  . finally , an important network metric is the degree - assortativity that tells us whether the hubs are directly connected in a network or if they are connected through intermediate nodes .we tabulate the statistical properties for the six networks in table 1 .we simulate the two epidemic models si and sir on these networks .si model helps us to understand diffusion and percolation in these networks whereas , sir is the classical model that describes epidemic spreading .both the models are simulated using agent based modelling technique ( igraph ) , where each node is considered to be an agent whose states ( s , i or r ) change with every increment in simulation .the si model is the most basic representation of an epidemic spreading model . in this model, there are two states that an agent or a node can exist in : s ( susceptible ) or i ( infected ) .the si model describes the status of individuals or agents switching from susceptible to infected at every instant of time .it is assumed that the population is homogeneous and closed , _i.e. _ , no new entity is either created due to birth or removed due to death , and also no new entity enters the system , thus preserving homogeneous mixing in the system .the si model also implies that each individual has the same probability to transfer disease , innovation or information to its neighbors .thus , the si model helps to capture the diffusion or percolation process in the entire network .the si model is formulated using the following differential equation . since an agent in the entire populationcan either be in state s or i , the si model is governed by a single parameter , , the infection transmission rate or simply , the infection rate .the growth in the number of agents in either of the sates is given by : substituting the value of s from equation ( 1 ) to equation ( 2 ) , we get the following differential equation describing the growth rate of i : the solution of the above equation with the initial condition at , is given by the logistic form :    contrary to the si model , the agents in sir model have access to three states s ( susceptible ) , i ( infected ) and r ( recovery ) . although the earlier assumptions of a closed population and homogeneous mixing also hold in this case , the complexity of the dynamical process increases due to the addition of one more state .the agents , instead of only switching between susceptible and infected ( as in si model ) , tend to recover in the sir epidemic model .the dynamics of the sir model is controlled by two parameters : the infection rate , , and the recovery rate , .the sir model can be mathematically represented by the set of the following differential equations : the population of susceptible nodes decrease in proportion to the number of encounters multiplied by the probability that each encounters results in an infection .the negative sign denotes that the population of s is decreasing .similarly , we can describe the evolution of the other two states , i and r. nodes become infected at a rate proportional to the number of encounters , and the probability of infection controlled by the parameter , .nodes recover at a rate proportional to the number of infected individuals , and the probability of recovery controlled by the parameter , : it would be interesting to analyze the spread of infection with respect to the susceptible individuals when there is a constant recovery ( from equation ( 6 ) ) .we calculate the variation in i with respect to s : the solution to the above equation with the initial conditions at , ( negligible as compared to the population ) and , is given by : in order to understand the rate of spread of infection in the population , we look at the rate equation for i from equation ( 6 ) : the above equation implies that the infection spreads if and only if . the epidemic dies out ( the number of infected individuals decreases ) if the above quantity is less than zero .bifurcation occurs at the stationary state , when , which separates the above two regimes and corresponds to the epidemic threshold ..tabular representation of the statistical data for the bus routes of six major indian cities ( = characteristic path length , = average weighted clustering coefficient , = power - law exponent , and = average node degree ) .[ cols="^,^,^,^,^,^,^,^",options="header " , ]   in figure 3 we simulate the si and sir models on the us airline network for 500 of the busiest airports ( )  .statistical analysis of the network reveals scale - free degree - distribution pattern between the nodes , with the characteristic path length and average degree .the si plot of the us airline network and abn show similar pattern of growth as both of them exhibit scale - free behaviour .however , the sir plot is similar to that of hbn due to extremely low characteristic path length and high average node degree . in table 2, we present our findings for the networks studied in the paper .the first column represents the simulation time ( in seconds ) for percolation threshold from the si model . in the second and third column, we present the epidemic thresholds for the various networks studied by computing the values of the plots from the sir model ( figure 2 ) ( as a fraction of network size ) and the corresponding simulation times ( in seconds ) respectively . in the final column, we present the characteristic path lengths for the various networks .finally , in figure 4 , we plot the variation in the rate of percolation by removing nodes from the network based upon their centralities and degrees . in transportation networks ,other than the degree of a node , closeness and betweenness centralities play a crucial role . while closeness centrality is a measure of a node s relative importance in the network due to the existence of shortest paths from that particular node to every other node in the entire network , betweenness centrality , on the other hand , acts as a bridging node connecting different parts of the network together . in order to capture their effects on information diffusionwe simulate the si model on modified networks generated after directed removal of nodes .since cbn and mbn are strongly assortative , we remove only two percent of the nodes ( higher number of node removal will cause cbn and mbn to disintegrate into disconnected components ) .we find that the removal of nodes does not significantly affect the diffusion in abn .however for cbn , dbn and hbn , we observe that when nodes are removed based upon their closeness centrality , the diffusion curve shifts towards right , thus signifying a delay in the diffusion process .this can be explained due to the fact that the removal of nodes based upon closeness centrality has a direct effect on the characteristic path length .a node with high closeness allows every other node in the network to be reached along the shortest paths .the removal of such a node affects / delays diffusion until the next central node is encountered . for mbn , we observe that degree - biased removal causes the diffusion rate to increase steeply , signifying the presence of redundant nodes that simply increase the characteristic path length of the network .a removal of of such nodes causes the diffusion to improve significantly , as can be compared from the simulation times recorded in figure 1 and 4 .in this paper , we simulated diffusion and epidemic spreading on the bus transportation networks for six different indian cities , and also compared the results with the us airline network . from table 2, we can clearly identify the characteristic path length to be a vital component in both , network diffusion and epidemic spreading .interestingly , the metric can be used as a single parameter to compare diffusion rates across different network topologies .this is because abn , dbn , hbn and kbn exhibit small - world phenomenon due to which the characteristic path length scales as the logarithm of the network size , _i.e. _ , . however , complexity arises when we have networks with comparable characteristic path lengths , like in the case of abn , dbn and kbn . in such cases, average node degree plays an important role . actually , a node having a degree has opportunities for infection . for a network with average node degree ,the rate of change of the susceptible population is given by : the rate of change of the infected and simultaneously recovered individuals is similarly given by : note that the rate of change of the recovered individuals remains the same as before .substituting the value of i from equation ( 11 ) into equation ( 10 ) , and solving for s gives us the following expression for s in terms of r , at time , and . population of recovered individuals , is given as , .recovery of the individuals occur in the network if and only if the slope of or .similarly , we can observe from table 2 that the values of epidemic thresholds ( dbn , hbn and us air ) are also strongly correlated with the characteristic path length .the epidemic threshold for dbn is particularly high compared to its comparable counterparts , abn and kbn .the reason for this behaviour lies in the fact that dbn has a high average node - degree when compared to abn , and is assortative when compared to kbn .this allows dbn a high degree of freedom for infection transmission .finally , the low epidemic threshold values for cbn and mbn can be directly attributed to their considerably higher magnitudes of characteristic path length . even though cbn and mbn are strongly assortative , the structural advantage of assortativity in diffusion is ruled out due to the presence of long routes mostly comprised of intermediate nodes .the effect of node centralities in information diffusion is also studied in this paper .it is found that networks are sensitive in percolation and diffusion to that particular metric which has a direct effect on the characteristic path length .for some networks , closeness plays a crucial role ( cbn , dbn and hbn ) .however for networks like mbn , degree - biased removal also reduces the magnitude of .although , the specific node centrality directly affecting the magnitude of largely depends upon the network structure , it can be argued that closeness of a node can be used as a marker for network immunization procedure .we studied the functionality of the bus networks of six major indian cities in this paper . since experiments with epidemic outbreaks in a population ( or a network ) is not a viable option , we resort to mathematical modelling .we , therefore , study the effect of percolation and epidemic spreading on these networks using si and sir epidemic models through numerical simulations . while it is observed that the characteristic path length plays a crucial role in information diffusion and epidemic spreading , several other network metrics also play important roles .their importance is however restricted to their relative contribution to the topological structures of the networks .small - world property , while an extremely desirable property in transportation networks , is highly subjective in its role of information diffusion , solely due to the diffusing entity .while diffusion of pathogens is an undesirable phenomenon , diffusion of useful information is a desirable one . in conclusion ,the work presented in this paper will help us in understanding and controlling the process of diffusion and epidemic outbreaks in spatially - constrained networks .the authors acknowledge the support from the information technology research academy , a division of media labs asia , a non - profit organization of the department of electronics and information technology , funded by the ministry of communications and information technology , the government of india and the center of excellence in urban transport at the indian institute of technology , madras , sponsored by the ministry of urban development , the government of india .   a contribution to the mathematical theory of epidemics . in _ proceedings of the royal society of londona : mathematical , physical and engineering sciences _ ( 1927 ) , vol . 115 , the royal society , pp . 700721

Abstract: bus transportation is considered as one of the most convenient and cheapest modes of public transportation in indian cities . due to their cost - effectiveness and wide reachability , they help a significant portion of the human population in cities to reach their destinations every day . although from a transportation point of view they have numerous advantages over other modes of public transportation , they also pose a serious threat of contagious diseases spreading throughout the city . the presence of numerous local spatial constraints makes the process and extent of epidemic spreading extremely difficult to predict . also , majority of the studies have focused on the contagion processes on scale - free network topologies whereas , spatially - constrained real - world networks such as bus networks exhibit a wide - spectrum of network topology . therefore , we aim in this study to understand this complex dynamical process of epidemic outbreak and information diffusion on the bus networks for six different indian cities using si and sir models . we identify epidemic thresholds for these networks which help us in controlling outbreaks by developing node - based immunization techniques .


people frequently face diverse situations that offer a wide choice of options , such as when looking for a restaurant , hotel , phone model or any basic good in the supermarket .the number of goods increases every day .it is estimated that approximately 50000 new products are introduced every year in the us  . even within each category of items, there may be many brands and item variations without differentiated attractiveness .this leads to the problem of facing too many choices , termed `` overchoice '' or choice overload  . to make decisions in such situations can be costly , and this stressful process leads to poor decisions or no decisions at all  , the advantages of multiple choices can be canceled by the disadvantages of a more complicated choice process .in fact , despite representing , apparently , a positive development , many options may hinder the process of choice .for example , people with many purchase options tend to have more difficulty in choosing and may end up buying nothing  .motivated by these observations , we wonder to what extent people interactions , leaving aside their individual psychology , contribute to this scenario by introducing , for instance , conflict and frustration .then , by means of a model of opinion dynamics , we investigate the distribution of adoptions made by a population facing a large number of choices .  in modeling people s interactions , one of the basic ingredients is imitation , or social contagion .in fact , imitation occurs in diverse social contexts , from the dynamics of language learning to decision making .depending on the questions posed , diverse rules of contagion , from simple pairwise to group interactions , have been proposed and studied in recent years  .however , very few works deal with many choices  . in the vast majority of opinion models ,the opinion of an agent is represented by a binary variable , since many questions can be tackled through the assumption of two possible ( opposite ) attitudes , e.g. , being either favorable or unfavorable to a given choice .this kind of binary variable was also inspired in the spin-1/2 ising model , leading to transpose known results from physical to social questions . for our present purpose of studying multiple - choice situations ,it is natural to consider a potts - like state variable that can take several ( discrete ) values .we consider that changes from one state to another are governed , not by simple pairwise contagion , like in refs . but , instead , by a `` plurality '' rule  .this is grounded on the idea that an individual makes the choice that is the most popular among its contacts .in fact , when we have to choose or buy something , especially when there are so many similar options that there is not a favorite one _ a priori _ and is not feasible to examine them all , it is reasonable to take into account other people s preferences  . naturally , the closer is the person in our network of contacts , more importance we give to its opinion , since nearest neighbors in the network typically have similar interests and tastes .one can use that strategy not only in trivial or daily life issues but also in major ones such as political elections where many candidates compete . for changing or adopting a new opinion, however , a minimum of consensus amongst the contacts is necessary .this is expressed in a plurality rule , according to which an individual is persuaded to adopt the opinion shared by the largest number of its nearest neighbors .evolution rules based on a locally dominant opinion have been considered before , for instance , the majority rule for two states introduced by galam  .it was later extended to multistate opinions  , by considering all - to - all interactions where the individual and all its contacts adopt the same opinion of the majority at the same time . a variant where , instead of the local majority , the plurality opinion is considered was also studied  .but in the situations we address here , the decisions are not taken in groups or simultaneously , rather the choices of the individuals are affected by their knowledge of the previous choices of theirs contacts .therefore , we assume that one single individual opinion changes at a time . moreover , we will include the possibility of undecided people , which is a realistic feature , which has been taken into account in 3-state models , as natural extensions of binary cases  .furthermore , as another differential , the dynamics of our model takes place in small - world networks , that , even if do not facilitate analytic treatment , are more realistic than regular or mean - field settings . as paradigms of small - world networks , we consider two classes with distinct degree distributions : erds - rnyi ( er )  and barabsi - albert ( ba )  networks .the details of the model will be defined in sec .[ sec : rules ] .the model encompasses instances where the different alternatives have similar initial attractiveness .many products sold in the internet with similar qualities and prices , e.g. , music albums , shoes , etc ., are within the model scope .we also make the simplification that individuals can differ only in the number of contacts .other heterogeneities of the agents might be also introduced in further work . by now, we ask a basic question : how a plurality rule molds the decision spectrum in the simplest , homogeneous , case ?first , we address the classical issue in this kind of problems , about whether a consensus state can be achieved or not , where all ( or almost all ) of the individuals share the same preference .then , in simulations of the model , we compute the fraction of the realizations reaching consensus . in non - consensus situations ,we analyze the distribution of adoptions .we also compute other relevant quantities such as the fraction of undecided people and the fraction of the population adopting the most popular alternative .the results will be presented in sec .[ sec : results ] .a plurality rule governs the opinion dynamics of agents interacting through their network of contacts .each agent , corresponding to a node in the network , has a potts - like opinion state variable , that can take the values representing electable alternatives ( options or choices , that we enumerate in an arbitrary order ) , as well as an `` undecided '' state , assessed when the individual has not adopted a defined option .the addition of the undecided state reflects the fact that sometimes people do not have a favorite choice .we focus on the dynamics developed in er and ba networks , as representative of small - world networks with homogeneous and heterogeneous degree distributions , respectively .however , for comparative purposes we will also consider random neighbors and nearest neighbors in a square lattice ( with periodic boundary conditions ) .we assume that most individuals do not have a formed opinion _ a priori _ , except for initiators representative of each offered choice .then , we start the dynamics with all nodes in the state , except randomly chosen nodes , to each of which we attribute a different opinion .we consider the same number of initiators ( one initiator ) for each alternative , reflecting the equivalent attractiveness of all the alternatives .this kind of initial state has been used in opinion models for proportional elections  .  at each monte carlo( mc ) step of the dynamics , we visit all the nodes of the network in a random order and update them successively , in asynchronous mode .the state of the visited node is updated according to the following steps :  a.  we define the set of nodes , , formed by and its nearest neighbors .b.  we determine the plurality state , associated to node , as the state shared by the largest number of nodes in the set .c.  the agent will then adopt its corresponding plurality state .notice that , when we measure the state , we ignore the nodes in that have ( see fig .[ fig : rules ] ) but the current opinion of site also counts to define .the updates are repeated and the dynamics stops when an absorbing state is attained , i.e. , if , at a mc step , none of the nodes changes its state .  , a node ( the central one in the figure ) and its neighbors form a group . in the case shown in the figure ,the plurality state is , since this is the state which is shared by more nodes .so , in the next time step , the node changes its opinion from to .notice that nodes with are ignored in the computation of the plurality state .]  let us remark that this dynamics differs from that of the sznajd type  where two or more individuals sharing the same opinion impose it to all their neighbors .it also differs in several aspects from the plurality rule introduced in ref . : i ) while in our case only the central node is affected , in  the whole group changes its opinion to the plurality state in a single update step , ii ) here the size of the interaction group is given by , where is the connectivity of site , instead of being constant ( anyway , the parameter plays the role of an effective , and they coincide in the limit of a highly homogeneous , or regular , network ) ; iii ) the possibility of indecision is not contemplated in  ; iv ) in an event of tie , the opinion of node remains unchanged in our model , while , in  , one of the dominant options is randomly selected .with respect to this last item , the present dynamics is more close to the majority rule version of ref . , where the dynamics becomes static because , when there is no local majority , the state of the group does not change .v ) in terms of the underlying network , here we consider small - world networks , while the dynamics in ref . was studied in the mean - field limit and over a square lattice .vi ) finally , another important difference is in the initial conditions : we consider that decided nodes are diluted in a sea of undecided nodes , instead of equiprobability of definite opinions .we follow the evolution of each realization of the dynamics until the final state is attained .distinct distributions of opinions can emerge in the final state depending on the amount of alternatives , the average connectivity , the network topology and size . fig .[ fig : dynamics ] illustrates the evolution of , the number of nodes that share the same opinion , in representative realizations of the dynamics on er networks of size .a final state is reached in a few monte - carlo iterations . fig .[ fig : dynamics ] illustrates the distinct patterns that can arise , while the full phenomenology as a function of the model parameter and network features will be shown in sec .[ sec : diagram ] .  , the number of nodes that share the same opinion , for an er network of size . for ,all opinions are shown , while for , only the winner opinion and other 20 opinions are shown .the winner opinion , as well as the opinion ( undecided state ) are highlighted .the level 50% of the population is represented by black dotted lines .the stationary state is attained in a few mc steps and distinct patterns arise for different values of and . ]\(a ) for instance , when and , we have a fragmented final state , where all initial values of have adopters .the number of undecided agents is predominant , representing about 20% of the population , in this particular realization .\(b ) for larger connectivity ( e.g. , ) , several opinions are still possible in the stationary state , but the winner opinion is widely dominant . the number of undecided agents in the steady state has decreased significantly with respect to case ( a ) .\(c ) for even larger connectivity ( e.g. , ) , consensus is likely .( d)-(f ) when is large , the number of undecided agents does not decrease monotonously when the connectivity increases .but , increasing the connectivity , consensus is reached , although it can take a larger time than for small .  in all cases ,the number of undecided nodes decreases with time , because undecided individuals are not produced by the dynamics in the present version of the model , but only introduced in the initial condition .the dynamics in er networks can be qualitatively understood as follows :  in a first regime , each opinion propagates invading the undecided neighbors .if the initiators are very diluted ( ) , and the connectivity is not too high , then each cluster of nodes with the same opinion can develop almost independently of each other , during several mc steps ( non - competitive regime ) . in this case , the initial growth is nearly exponential , described approximately by in er networks .when two or more clusters collide , a competitive regime starts .depending on the network , the competition can take place more or less evenly so that ties stagnate the evolution avoiding wide dominance of a given opinion ( as in fig .[ fig : dynamics].a , [ fig : dynamics].d and [ fig : dynamics].e ) .otherwise , a sort of rich - get - richer or cumulative advantage mechanism can take place . in that case, the winner opinion becomes noticeably larger than other ones , convincing individuals from other opinion clusters ( fig .[ fig : dynamics].b ) or even the whole network ( fig .[ fig : dynamics].c ) .similar patterns as those shown for er networks are also observed for ba networks , although for different values of the parameters , as illustrated in the first column of fig .[ fig : dynamics2 ] .notice that the winner opinion , as well as the number of decided people , for the same parameters , are favored in ba networks , where cumulative advantage effects are more accentuated .let us remark that in some extreme situations , given the initial conditions studied here ( one initiator for each state ) , the system does not evolve .this occurs , for instance , in the limit case when ( hence , each individual has a defined opinion ) or the connectivity of all sites is ( complete graph ) . in those extreme cases , ties forbid changes of state and the dynamics is frozen from the start .however , we will restrict the range of the parameters to the region ., with the same conventions of fig .[ fig : dynamics ] , for ba networks , square lattices ( sq ) and mean - field ( mf ) interactions ( random networks ) for the parameters indicated in the figure . in panels ( b ) and( e ) the purple dashed lines are given by eq .[ sq ] . in all cases . ]  aside from networks with the small - world property ( er and ba ) , we also analyzed , for comparison , interactions with ( i ) first neighbors in square ( sq ) lattices ( with periodic boundary conditions ) , and ( ii ) random neighbors ( with ) .representative examples of the evolution , in square lattices and random neighbors for the case , are shown in the second and third columns of fig .[ fig : dynamics2 ] .  in regular lattices ,the evolution is essentially non - competitive .clusters grow from their initiators and , when they collide , the dynamics freezes due to ties in the interfaces , without entering a competitive phase , in contrast to er and ba networks where there are long - range links that break ties . since the initial growth occurs at the surface of the cluster ,then which , differently from small - world networks , gives a quadratic increase of with , as observed in fig .[ fig : dynamics2 ] ( second column ) .alternatively , it is easy to show that , for the synchronized update , at short times , the number of adopters of each choice grows around its initiator , following , in average , the recursion relation , where for the square lattice . by solving this recursion equation ,one obtains ,\ ] ] that yields the predicted quadratic increase with , valid for small , until holds . despite the predictionis done for the synchronous update , it is in good agreement with the average value of the simulated curves , as can be seen in the second column of fig .[ fig : dynamics2 ] .  as a consequence of the lack of competition , the final values of are lessdisperse in the square lattice than in random networks and , mainly , consensus , or even a wide dominance of an opinion , becomes unlikely for .  in the absence of any network structure ( i.e. , when neighbors are chosen purely randomly ) , like in the examples of the last column of fig .[ fig : dynamics2 ] , one of the opinions dominates and attains consensus , even for very small .this can be understood in terms of a mean - field approach , following the lines of ref .in fact , the fraction of undecided sites follows an equation of the type where is a polynomial of order in whose coefficients depend on the fractions , with and . for all and , the factor arises from the central undecided node that is part of the group .therefore , in the steady state it must be .moreover , since , eq . ( [ zero ] ) is stable for .the remaining equations for the other fractions have solutions of the type found in ref . for their majority and plurality versions .in particular , the stable solutions are those of consensus , where =1 for some ( hence , the remaining fractions vanish ) .  for instance , for and ( let us call , ), we have \ , , \nonumber \\ \dot f_1 & = &  f_0f_1(2f_0 + f_1 ) + f_1f_2(f_1-f_2 ) \ , , \nonumber \\   % f_0  f_1  f_2 \dot f_2 & = &  f_0f_2(2f_0 + f_2 ) + f_1f_2(f_2-f_1 ) \ , .\nonumber\end{aligned}\ ] ] it is easy to obtain that the only stable solutions are and .  similarly , for and , once , we have that lead to the consensus solutions as the stable ones , while equipartition is unstable . the equationsabove , for small values of and , valid for in the present model , are the same obtained for the majority and plurality versions studied in ref . , although the equations in three cases differ for enough large values of and .increasing leads to equations of the form , where is a definite positive polynomial of order in , whose coefficients depend on . therefore , consensus is always stable for .  for large number of alternatives, necessarily must vanish as well , and consensus is also a stable solution .for instance when and , once , we have that lead to the consensus solutions as the stable ones , while equipartition is unstable , and the solutions of the type are saddle points .increasing and , the structure of fixed points becomes more complex and more routes to consensus emerge  .nonetheless , consensus is always the final state , which was also verified through numerical simulations .differently , when the structure of the interaction network is relevant , non - trivial behaviors occur , as those illustrated in fig .[ fig : dynamics ] .a population of undecided people can survive and consensus is not always attained .   to summarize , the nontrivial final configurations that emerge in er and ba networks , we built a phase diagram in the plane .for each realization , we monitored the fraction of decided people and also the fraction of nodes sharing the most adopted opinion , or winner choice ,   these quantities were averaged at the final state of several realizations .unless said something different , at least 50 realizations were considered for each set of values of the parameters . for each realization of the dynamics ,a different network was generated .the averaged fractions will be denoted by and , respectively .we also computed the fraction of the simulations that reach consensus ( operationally meaning at least 99% of the population ) .  when performing computations over er networks , only the main component of the graph was considered .   showing the fraction of consensus in color scale .full symbols emphasize the points where consensus occurs in all ( , dark red ) and none ( , dark blue ) of the 30 realizations for each cell , otherwise symbols are empty .the solid lines depict the points such that ( to the right of the curves ) .the shadowed areas represent the points for which .er and ba networks size is . ]the phase diagram in the plane of parameters , for er and ba networks is depicted in fig .[ fig : diagram ] .we restricted the analysis to the region and . the diagram shows the changeover from regions of consensus to regions of fragmented final state , as indicated by the colors from red to blue .filled symbols emphasize the points where consensus is certain ( , red ) or uncertain ( , blue ) .moreover , the solid lines depict the frontier where ( to the right of the curves ) , and the shadowed areas highlight the points for which undecided people are majority ( ) .it is clear that the consensus domain ( , red region ) is larger for ba networks , pointing out that heterogeneity favors consensus . in both er and ba networks , inserting a few links , for fixed near the transition frontier , may trigger consensus .  in ba networks , for sufficiently large ( )the critical value becomes independent of .in contrast , in er networks , the dependence on is stronger .the non - consensus domain for er networks , ( , blue region ) , when becomes sufficiently large , spreads over the region of large mean connectivities .it means , that near the transition frontier , eliminating a few alternatives can trigger consensus , an effect which in ba networks only occurs for connectivities below .  concomitantly , in the shadowed area in fig .[ fig : diagram].a ., the fraction of decided people becomes minority ( ) .differently , in the ba case , the majority of nodes is decided over the whole phase diagram . moreover , even in the absence of consensus , the winner group can become majority ( ) , more easily in ba networks .  in the following sections [ sec : q ] and [ sec : k ] , we describe in more detail the dependency on and , respectively .   in this section ,we focus on the impact of the number of opinions on the steady state , and we also discuss size effects .  fig .[ fig : opinions ] shows the three quantities of interest , , and , as a function of , for er and ba networks and sq lattices of three different sizes .the plots of vs , for er and ba networks , correspond to vertical cuts of the phase diagram in fig .[ fig : diagram ] .all the fractions monotonically decrease with , in the studied range , indicating that independently of the lattice , increasing the number of options hinder the process of decision making , promoting indecision , and also turning more difficult the appearance of a popular choice .concerning size effects , in er networks ( panel a ) , the three quantities are almost independent of size .the results in square lattices ( panel c ) are in accord with the predictions made in sec .[ sec : dynamics ] .for instance , only for .the winner fraction is independent of and decays subtly above , as expected .in fact , we have seen that the fractions of each opinion have a narrow distribution around the mean value .meanwhile , the decided fraction depends on .since undecided sites are at the interfaces of each opinion cluster , then , their quantity over all clusters is .therefore , the decided fraction results as a consequence , the curves of for different sizes collapse when represented vs , as shown in the inset of fig .[ fig : opinions].c .in contrast , in ba networks ( panel b ) , and also depend on .moreover , does not vanish but tends to a finite value as increases .the heterogeneity of these networks favors that the winner conquers a large domain , which increases with system size .the decided fraction also depends on and data approximately collapse when represented vs , with for the case of fig .[ fig : opinions].b ( see inset ) .this value of suggests that the number of `` interfacial '' nodes scales with the number of bulk nodes in a cluster with exponent .the fraction , as well as the critical value at which vanishes , follow the same scaling , as shown in the inset of fig .[ fig : opinions].b .( triangles ) , average fraction of the winner choice ( squares ) , and fraction of realizations reaching consensus ( circles ) , as a function of the number of different opinions .three different sizes were used for er and ba networks with mean connectivity , and for square lattices .the number of nodes is given for networks and the linear size for lattices .solid lines are a guide to the eye , and shadowed regions represent the standard deviations of and in the case ( that are similar for the other sizes ) . in panel( c ) , the dashed line corresponds to .computations were done over at least 50 samples .the insets show the data collapse , as explained in the text : in ( b ) , ; in ( c ) , the dashed line is given by eq .( [ fdsq ] ) . ]the average fraction of decided people typically decreases with ( for ) , but a kind of saturation effect occurs for large enough and a flat level appears in small - world networks , indicating that becomes insensitive to the introduction of new choices .however , the value of the flat level changes with the network connectivity ( not shown ) . for large number of options ,the values of are smaller in er networks .heterogeneity of degrees seems to be helpful in breaking ties . in the heterogeneous ba networks , many nodes have low connectivity andcan be easily convinced by a decided neighbor . on the other hand , the occurrence of a local plurality is less likely in an homogeneous er network with a given connectivity . as a consequence ,ties are more frequent , more nodes remain undecided and the dynamics freezes . however , in square lattices , despite the homogeneity, the undecided fraction is relatively small .this can be understood as follows .ties occur when distinct opinion clusters collide , then , the surviving undecided nodes are located at the `` interfaces '' . in networks with long - range linksthe encounter of different clusters occurs early , and many nodes remain undecided , meanwhile in regular lattices with nearest - neighbor interactions , undecided nodes are being conquered until the collision , late in the dynamics , when the opinion groups have occupied most of the lattice and few undecided nodes remain at the interfaces .the average fraction of the population adopting the winner option , ( squares ) is also a significant quantity .( necessarily . )the fraction is greater in ba networks .that is , the winner choice conquers in average a large fraction of the population in ba networks , compared to er networks and square lattices with equivalent .in fact , the cumulative advantage that drives the growth of an opinion group is facilitated in these heterogeneous networks due to the presence of hubs , and the winner conquers more adopters .also notice that , in er networks , when the fraction attains the flat level , a dominant opinion is absent , as mirrored by the very small value of ( see fig .[ fig : opinions].a ) .meanwhile , in ba networks the winner can always conquer an important fraction of the population ( fig .[ fig : opinions].b ) , shown by the fact that remains finite ( except in the limit ).  for all kinds of networks , the probability of occurrence of consensus , ( circles ) , typically falls from 1 to 0 as increases .this agrees with the intuition that , when there are more options to choose , it is more difficult to attain consensus .the probability of consensus decays rapidly with and , above a critical value , the fraction becomes negligibly small .this effect is accentuated in the square lattice where for .the behavior of the characteristic fractions , and as a function of the average connectivity of the network , , are shown in fig . [fig : connectivity ] , for several values of , in er and ba networks .the plots of vs correspond to horizontal cuts of the phase diagram in fig .[ fig : diagram ] .let us start by the case of ba networks ( shown in fig .[ fig : connectivity].b ) that exhibits a simple monotonic behavior , for the range shown in the figure .the three fractions increase with the connectivity . as observed in the phase diagram of fig .[ fig : diagram].b , fig .[ fig : connectivity].b shows in more detail how the jump to consensus becomes more abrupt as increases and the critical value of the connectivity becomes nearly independent of the number of options ( ) , as also observed in the phase diagram .differently , in er networks ( see fig . [fig : connectivity].a ) , increases with .moreover , the average fraction of decided people first decreases with the connectivity down to a minimal value localized at .up to that point , the fraction of simulations attaining consensus is negligibly small .but , at , a transition occurs and both and rapidly increase with , up to 1 .one would expect to have more decided nodes when the connectivity is higher , like in the case of ba networks , since , in principle , more connections might facilitate information spreading .however , on a low connected network , opinion groups are typically isolated from each other . when links are added , and disconnected groups become connected , ties can occur .that is , on the one hand higher connectivity implies that groups of different opinions can be more connected among them and compete . on the other, a node will be aware of more opinions , making difficult the decision and keeping more undecided nodes .therefore , not only overchoice ( high ) may produce stagnation of the dynamics but also `` overlink '' or excess of contacts due to high .this explains the initial decrease of with which occurs up to a minimal value of .after that point , introducing more connections will allow a dominant group to impose its opinion , concomitantly increases until reaching its maximal value 1 .also notice that in er networks , for large values of , there is an interval of mean connectivity for which the fraction of decided people becomes minority .( full lines ) and ( dotted lines ) as a function of the mean connectivity , for several values of , indicated by different colors , in er ( a ) and ba ( b ) networks of size . for the sake of clearness ,the behavior of the winner fraction ( black dotted lines ) is depicted only for the case .averages where computed over at least 50 samples . ]the existence of an abrupt transition from a situation where many opinions coexist to consensus indicates that , by adding just a few links or by removing a few choices , most people may come to adopt the same state .the transition to consensus is more abrupt in ba networks , and the jump width decreases with . in these networks , as discussed above , there is a dominant winner opinion group , that represents an important fraction of the population ( finite ) .the largest group gains additional adopters more easily , with cumulative advantage . near the critical connectivity ,when adding few links at random , it would be more probable to connect the very large group to smaller ones , and , as a consequence , they would be conquered by the dominant opinion , rapidly leading to consensus . in homogeneous er networksthis transition is less abrupt , because a largely dominant group is less probable .the width of the transition region slightly increases with . for the extreme case of nearest neighbors interactions in square lattices , cumulative effectsare completely absent , therefore , a transition to consensus is unlikely .   in non - consensus steady states ,a broad distribution of opinions across the population can emerge . in order to analyze the shape of the distributions, we built the normalized histograms of , where is the number of nodes with a given opinion .histograms were computed by accumulating realizations ending in non - consensus states .typical distributions are depicted in fig .[ fig : pdfs_k ] .one can identify exponential , log - normal and power - law behaviors ., , built over 50 samples , for , on er ( a ) and ba ( b ) networks with size , and different values of indicated on the figure .the solid lines are log - normal fits , the dotted lines exponential fits , and the dashed line with slope -3 was drawn for comparison . ]  in the er case , far enough from the critical frontier of consensus , the preferences are almost uniformly distributed with an exponential cutoff .when approaching consensus ( for instance by increasing ) the distribution adopts a log - normal shape .notice that this occurs in the region of the phase diagram where indecision prevails ( shadowed area in fig .[ fig : diagram ] ) . in ba networks ,the distribution can also resemble a log - normal , but when approaching consensus the tail rises due to the existence of dominant winners .moreover , when the dynamics freezes early , tends to reflect the degree distribution with exponent ., of the number of ( favorable ) reviews , with 4 and 5 stars , for netshoes  and google play music  .it is reasonable to identify the number of items with , and the number of reviews giving 4 and 5 stars with the number of decided people . for google play musics , prices are similar ( u 200 ) . the solid lines are log - normal fits . ]  in order to compare the distributions from simulations with those from real world , we considered products than are rated online .we analyzed data about items whose alternatives are not significantly differentiated ( for instance , in price and/or quality ) , as assumed in our model .we identified with the number of items within each category and we considered the amount of positive reviews ( those of 4 and 5 stars ) received by each item as indicator of its total number of adopters , that potentially might become spreaders of the product , in a situation alike that described by the present model .we analyzed all music albums from google play music  , whose prices are similar ( u ) . for each set of data we computed the histogram of the number of adoptions ( i.e. , number of reviews attributing 4 and 5 stars ) , as shown in fig .[ fig : pdfs - real ] .comparison of figs .[ fig : pdfs_k ] and [ fig : pdfs - real ] put into evidence a remarkable qualitative similarity between real and simulated distributions producing log - normal shapes .the shape for small values of is also similar .once service users have access to the reviews of any other user , the underlying network is expected to be similar to a random graph with relatively high connectivity .since purely random ( mean - field ) interactions would lead to consensus , which is not observed in empirical data , one concludes that the underlying network must have some structure .the absence of a fat tail , related to the presence of hubs , as in ba networks , indicates that the empirical cases are best modeled by er networks , at least qualitatively . in fact , in a `` rating network '' , reviews are equivalent and none of them is expected to act as a hub , then , it is reasonable that er networks yield more realistic results in this case .we introduced a model based on a plurality rule , that mimics decision making governed by the influence of the relative majority of the neighbors .the model also incorporates the possibility of undecided agents .it applies not only to situations involving consume of products or services but also to other environments where there is a variety of options , as far as the options are homogeneous with similar attractiveness ( similar quality , cost , etc . )and people have no preferred choice a priori .different final steady states emerge from the dynamics , depending on the number of available options and on the degree distribution of the network of contacts : consensus , wide distribution of opinions or also situations where indecision dominates for sufficiently large number of options .in fact , decision making governed by the plurality rule may yield ties ( conflict and frustration ) , contributing to overchoice stagnation .this effect appears to be mitigated in ba networks .the model envisages that stagnation may result not only from overchoice but also from the excess of links ( see fig .[ fig : connectivity].a ) .  for both types of small - world networks ,consensus is almost certain for sufficiently low number of options and sufficiently large connectivity . for er networks, consensus can occur even if there are many options available . if neighbors are random , consensus is the rule .but consensus is unlikely for large number of options and low network connectivity , specially if the network is homogeneous . in the square lattice with periodic boundary conditions consensusis not reached , but opinions tend to be equipartitioned . in small - world networks , there are nontrivial non - consensus regimes , and a broad distribution of opinions can emerge , with a shape similar to that of real ones when the assumptions of homogeneity hold , like in the examples of netshoes and google music albums .the model indicates that consensus can suddenly emerge simply by introducing a few connections or eliminating a few items .furthermore , it also predicts that an item can become very popular ( with relatively large ) , even if the initial attractiveness of all the items is uniform .these observations furnish another possible explanation of why there is so much amateur content viralizing in the internet , or why a service , good or cultural product can become a bestseller without having any apparent differentiated attractiveness .we acknowledge financial support from brazilian agencies cnpq , capes and faperj .40ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] ,  _ _ ( , ) http://www.chicagobooth.edu/research/workshops/marketing/archive/workshoppapers/vohs.pdf [ ( ) ] * * ,  ( ) __ , harper perennial ( , ) * * ,  ( ) link:\doibase 10.1016/0022 - 2496(86)90019 - 2 [ * * ,  ( ) ] link:\doibase 10.1002/ejsp.2420210105 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) http://arxiv.org/abs/cond-mat/9702163 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) _ _ ( , ) http://dx.doi.org/10.1088/1742-5468/2006/03/p03013 [ ( ) ] * * ,  ( ) http://dx.doi.org/10.1088/0305-4470/38/33/003 [ * * ,  ( ) ] * * ,  ( ) http://link.springer.com/article/10.1140/epjb/e20020045 [ * * , ] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0136083 [ * * ,  ( ) ] * * , ( ) http://dx.doi.org/10.1209/0295-5075/78/18002 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) http://science.sciencemag.org/content/286/5439/509 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) * * ,  ( )

Abstract: people are often challenged to select one among several alternatives . this situation is present not only in decisions about complex issues , e.g. , political or academic choices , but also about trivial ones , as in daily purchases at a supermarket . we tackle this scenario by means of the tools of statistical mechanics . following this approach , we introduce and analyze a model of opinion dynamics , using a potts - like state variable to represent the multiple choices , including the `` undecided state '' , that represents the individuals that do not make a choice . we investigate the dynamics over erds - rnyi and barabsi - albert networks , two paradigmatic classes with the small - world property , and we show the impact of the type of network on the opinion dynamics . depending on the number of available options and on the degree distribution of the network of contacts , different final steady states are accessible : from a wide distribution of choices to a state where a given option largely dominates . the abrupt transition between them is consistent with the sudden viral dominance of a given option over many similar ones . moreover , the probability distributions produced by the model are validated by real data . finally , we show that the model also contemplates the real situation of overchoice , where a large number of similar alternatives makes the choice process harder and indecision prevail .


recently , significant amount of digital data on the behavior of individuals has enabled us to quantitatively explore the various patterns of human dynamics .one of the robust findings is that human dynamics is not random but correlated , such that the bursts of rapidly occuring events alternate with long inactive periods  .the bursty dynamics is typically characterized by the heavy tailed or power - law distributions of waiting time or inter - event time as with and for e - mail and mobile phone call ( mpc ) communications  , respectively .the origin of bursts in human dynamics has been debated for the past few years .the bursts can emerge from the individual selection of tasks from the task list according to the protocol `` the highest priority task first ''  .the long waiting time of a task with low priority leads to the heavy tail of the waiting time distribution .this model assumes that there are at least two tasks to be compared , which is not necessarily the case in reality . on the other hand the circadian and weekly activity patterns of humans can also result in the long periods of inactivity that contribute to the heavy tails  . in this approach the circadian and weekly patternshave been modeled by the non - homogeneous poisson process , implying that human behavior is essentially random .while both models are based on intuitive and to some extent realistic assumptions , they are not able to consider other human related factors properly , such as how individual agents reply to the received e - mail or phone call , and what is the benefit or cost of communication .  in order to investigate the role of human factors in bursty human dynamics we devise a model , where an individual agent reacts upon the lack of information , such that an information seeking agent sends an e - mail or makes a phone call to an information provider . with this modelwe describe the time frame of how long people wait for responses from the information providers under uncertain ( risky ) situations and get insight into which human factors in the daily communication are relevant to the bursty dynamics . herewe start with an assumption that an agent prefers a fixed amount of payoff over a risky lottery of the same expected payoff , which is called _ risk - aversion_. a risk - averse agent can reduce uncertainty by communicating for information , which is a time - consuming process .once an agent requests information , we consider the time to wait for responses , i.e. the waiting time , as a cost .this reflects a well - known economic perspective that time and information are considered as tradable goods  .based on the trade - off between the information gain and the cost of the waiting time , the agent chooses an optimal waiting time .note that we confine our model to risk - averse agents because the optimal waiting time for risk - pursuing agents is always zero and the waiting time does not matter for risk - neutral agents unless the information gain is unequal to the cost of time . by optimizing the waiting time, we derive a specific ( positive ) relationship between the risk and the optimal waiting time .then , we show that the optimal waiting time distribution follows a power - law , which serves as theoretical support for the observed bursty dynamics in e - mail and mobile phone call communications .our model gives insight into the human communication dynamics , i.e. how people are concerned with the communication efficiency and the cost per unit time while deciding the communication channel as well as the time to wait for responses .since the efficiency and the cost per unit time are heterogenous in different communication channels , the optimal channel depends on how urgent the situations are and how patient the users are .furthermore , the efficiency and the cost per unit time also change the optimal choice of the waiting time .we find that if the cost per unit time dominates over the efficiency , people are concerned with both the efficiency and the cost per unit time under the low risk situation , while they consider only the efficiency under the high risk situation .this explains the effect of human factors on communication such that in riskier situation people tend to focus more on reducing the risk without concerning the cost , because shortening the waiting time by the efficient communication is preferred to concern the cost itself .however , if the cost per unit time is dominated by the efficiency , the result is opposite .people are concerned with only the efficiency under the low risk situation and consider both the efficiency and the cost per unit time under the high risk situation , which implies that people do not consider the cost per unit time seriously under the low risk ( short waiting time ) but do under the high risk ( long waiting time ) . in conclusionthe individual agents attitude towards risk can be one of the explanations for the pattern of the waiting time in communication .in section [ sect : model ] we describe and analyze the model and explain the implications of the results .then we make conclusions in section [ sect : conclusion ] .as in  we consider the risk - averse utility function , of state , whose degree of risk - aversion is measured by . for simplicity , we set and then consider a risk - averse agent with utility function given by corresponding to a time - dependent state . at the beginning , the state is uniformly distributed from to .then , after waiting time for information about the state , the agent reduces the uncertainty on the state as .that is , the state becomes more specified .the decreasing speed of uncertainty is controlled by the parameter , which we call _efficiency_.  as an example let us consider a job applicant who got offers from 5 firms , indexed by and let be the set of wages of firms , which is the initial `` state '' if the applicant is uninformed .he chooses one of them and his utility depends on the wage of the firm chosen such that .then suppose that the applicant can obtain information on the unknown wages .if the information he will get at time is the exact amount of , the state becomes . at time , the applicant can make a clearer decision than he could at time because the number of unknown wages at is 4 . as the applicant knows the wages one - by - one ,his decision gets improved , and eventually at time , he will know the firm of the highest wage so that the best decision without uncertainty can be made .the time the agent waits for information to reduce the uncertainty can be considered as a cost .let be the cost of time . for simplicity , we assume that the cost per unit time is constant but dependent on the uncertainty level such that with positive and .this assumption describes that given the level of the uncertainty , the cost of time is constantly increasing in time at the rate of uncertainty except for the case with .the larger value of corresponds to the larger cost of the time unit , indicating that the parameter controls _ the cost per unit time_.  now we define the agent s expected utility at state subtracted by the cost of time as follows : -c(t)\nonumber\\   & = & a -\int_{-\infty}^\infty dx \sqrt{\frac{t^\gamma}{2\pi\sigma^2 } } \exp \left(-x -\frac{x^2t^\gamma}{2\sigma^2}\right)-c(t)\nonumber\\   & = & a - \exp\left(\frac{\sigma^2}{2t^\gamma}\right ) -c(t).\label{eq : expectutil}\end{aligned}\ ] ] the agent chooses the optimal waiting time for information . at maximizing eq .( [ eq : expectutil ] ) , the following condition is satisfied : from this equation and by means of lambert function , satisfying , we obtain the optimal waiting time as a function of , as follows }\right)^{-1/\gamma},\label{eq : tau_exact}\ ] ] where ^{1/\gamma} ] .therefore , we find two scaling regimes as for and for , respectively . here denotes the crossover uncertainty and also defines the crossover waiting time .it indicates that the optimal waiting time distribution shows two scaling regimes with power - law exponents for and for , respectively .note that for any positive and .  intuitively , the optimal behavior of individual agents is changing according to the risk . under the low risk situation ( ) ,the power - law exponent as a function of and explains the case where people are concerned with the efficiency as well as the cost per unit time when the risk is not significantly high .on the other hand , under the high risk situation ( ) , the power - law exponent turns out to be independent of . unlike for the low risk situations ,people under the high risk consider only the efficiency , i.e. it is better to obtain information as quickly as possible due to the high cost of time .note that the special condition of yields , i.e. one scaling regime of optimal waiting time distributions .[ cols="^ " , ]   we investigate the full range of eq .( [ eq : tau_exact ] ) by numerically solving for various values of , , and .the effect of constant is systemically considered by scaling and as and } & \textrm{if ,}\\    & \textrm{if ,}\\   \end{tabular}\right .\label{eq : largebeta}\ ] ] where }$ ] .the power - law exponent for the low risk regime , , turns out to depend only on , while the power - law exponent for the high risk regime , , is a function of both and .in other words , when the cost per unit time is dominated by the efficiency , people are concerned with only the efficiency under the low risk situation .this is because under the low risk , implying short waiting times , the cost of waiting time is still negligible compared to efficiency . on the other hand ,people under the high risk are concerned with the efficiency as well as on the cost per unit time because the cost of time is not negligible any more due to the long waiting time .  for the full range of , we have numerically solved the eq .( [ eq : tau_exact ] ) . in case with , implying , we obtain the numerical solutions for and and for , , and . as shown in fig .[ fig : beta](b ) , for each value of , the curves for different values of collapse to a single curve of eq .( [ eq : universal ] ) with two scaling regimes .in order to investigate the role of human factors in the bursty dynamics of individuals , we have studied an agent - based model .our model assumed an intrinsic human behavior , such as a trade - off between information gain to avoid uncertainty and the cost of time to wait for information . with the agent s optimal choice for the waiting time , the empirical waiting time distributions in daily communication process have been explained .the results we obtained by analytical derivation show two scaling regimes for the waiting time distributions with different power - law exponents , denoted by .this indicates that people have different attitudes on valuing the efficiency and the cost per unit time , controlled by and , respectively .while is generally expected to be a function of both and , the value of turns out to be independent of in two cases : a ) when and under high risk situation , people are not concerned with the cost per unit time because it is better to obtain information as quickly as possible due to the high cost of waiting time .b ) when and under low risk situation , the cost of waiting time becomes negligible so that people do not have to be concerned with the cost of time unless the waiting time is significantly long .we showed that the scaling behavior of optimal waiting time distributions reveals which human factors in the daily communication are considered as relevant to scaling .we also expect that further studies to identify intrinsic properties of social agents as well as extension of this model to interacting agents on complex networks would be helpful in figuring out the origin of bursts in human dynamics even in more details .  financial support by aalto university postdoctoral program ( hj ) , from eu s fp7 fet - open to ictecollective project no. 238597 , and by the academy of finland , the finnish center of excellence program 2006 - 2011 , project no .129670 ( kk ) are gratefully acknowledged .

Abstract: human dynamics is known to be inhomogeneous and bursty but the detailed understanding of the role of human factors in bursty dynamics is still lacking . in order to investigate their role we devise an agent - based model , where an agent in an uncertain situation tries to reduce the uncertainty by communicating with information providers while having to wait time for responses . here the waiting time can be considered as cost . we show that the optimal choice of the waiting time under uncertainty gives rise to the bursty dynamics , characterized by the heavy - tailed distribution of optimal waiting time . we find that in all cases the efficiency for communication is relevant to the scaling behavior of the optimal waiting time distribution . on the other hand the cost turns out in some cases to be irrelevant depending on the degree of uncertainty and efficiency .