Modelling indirect interactions during failure spreading in a project activity network

Spreading broadly refers to the notion of an entity propagating throughout a networked system via its interacting components. Evidence of its ubiquity and severity can be seen in a range of phenomena, from disease epidemics to financial systemic risk. In order to understand the dynamics of these critical phenomena, computational models map the probability of propagation as a function of direct exposure, typically in the form of pairwise interactions between components. By doing so, the important role of indirect interactions remains unexplored. In response, we develop a simple model that accounts for the effect of both direct and subsequent exposure, which we deploy in the novel context of failure propagation within a real-world engineering project. We show that subsequent exposure has a significant effect in key aspects, including the: (a) final spreading event size, (b) propagation rate, and (c) spreading event structure. In addition, we demonstrate the existence of ‘hidden influentials’ in large-scale spreading events, and evaluate the role of direct and subsequent exposure in their emergence. Given the evidence of the importance of subsequent exposure, our findings offer new insight on particular aspects that need to be included when modelling network dynamics in general, and spreading processes specifically.


Introduction
Recent years have witnessed a flurry of work on spreading processes 1,2 , ranging from empirical expositions on the impact of such spreading (e.g.existence of large-scale spreading events [3][4][5] , properties of 'super-spreaders' 6-10 ) to methodological developments that map the underlying dynamics (e.g.spreading mechanisms [11][12][13][14] , modelling frameworks [15][16][17][18][19] ).Pairwise interactions are the perceived centrepiece in understanding the evolution of spreading processes 20 , since they capture the direct exposure of each node to the prospect of switching its state, for example from 'non-affected' to 'affected'.Importantly, the impact of direct exposures can be supplemented by indirect exposures which emerge from local, non-trivial structures within the network (e.g.particular network motifs such as the feed-forward loop 21,22 ).Despite the intuitive importance of these indirect exposures 21,22, , little attention has been paid in evaluating their impact to the overall spreading process, largely due to the particularities of the spreading models typically deployed to study these processes.
In particular, spreading models can be classified to two broad classes 12,23 , depending on the incorporated mechanisms: (i) epidemiological models, where the spreading process is viewed as an independent event across different pairs of nodes (e.g. the probability of node  to affect its neighbour  is independent of other interactions), and (ii) sociological models, where the spreading process is viewed as a locally-dependent event (e.g. the probability of node  to affect its neighbour  depends on the state of node's  neighbours).Despite this distinction, both model classes are grounded on the same fundamental premise, in which direct exposures are the principal determining factor that determines the dynamics of the spread.Yet, the implication of this premise can be non-trivial, as indirect exposures can interfere with the spreading process and affect key outcomes, even in simple examples like the one discussed below.
Consider the toy network in Fig. 1, where each node can switch states irreversibly from 'non-affected' to 'affected' with a given probability .In the case of an epidemiological spreading model, this premise corresponds to the widely used 'Susceptible'-'Infected' model 24 , where the ability of node  to affect its immediate neighbours is assessed independently across all possible pairs (pair  → ; pair  → ; pair  → ).At this point, node  is directly exposed to the failure of node , through the directed link from node  to node , and indirectly, through paths  →  →  and  →  → .As a result, the possibility for node  to switch state is evaluated at three distinct points during the evolution of the spread, compared to the single evaluation that takes place in the case of node  and node .Hence, node  is three times more likely to switch state, compared to node  and , despite that fact that the probability of them changing state is uniformly set.We can trace this effect to the implicit assumption that direct and indirect exposures are of equal importance, which is appropriate when dealing with a typical disease spreading.
Fleshing out this assumption, the probability of node  to switch states, either due to its contact with node , or due to its contact with node  (as part of path  →  → ), is exactly the same.This suggests that the underlying pathogen that drives the spread has remained unchanged (and therefore, the probability for node  to infect its neighbours, including node , is exactly the same the as the probability of node  to infect its own neighbours, including node ).However, consider an alternative context where the pathogen is replaced by a defect, which spreads across a network of activities, where nodes correspond to activities and links to functional dependencies 25,26 .In this case, the interpretation of direct and indirect exposures is distinct, where activity  may be immune to the particular defect that caused activity  to fail.Therefore, the fact that activity  is additionally exposed to the defect in an indirect way (through node  and node ; Fig. 1) should not interfere with the likelihood of activity  changing its state; yet typical spreading modelslike the aforementioned SI modelwould assume so.In response, we propose a simple spreading model that allows us to map the impact of direct and indirect exposures.It does so by disentangling the probability of a defect to spread  to two distinct components:  1 , which controls the probability of node  being affected by the failure of its predecessor, node  (Fig. 1, top panel) and  ∞ , which controls the probability of node  being affected by additional, indirect exposures to the failure of node , through node  (Fig. 1, bottom panel) or node  (Fig. 1, bottom panel).As such, we formulate two models,  0 where  1 ∈ [0,1] and  ∞ = 0, and  where both  1 ∈ [0,1] and  ∞ ∈ [0,1].Hence, any difference between results obtained under  0 and  reflect the impact of indirect exposure, with  ∞ controlling the magnitude of the effect.We deploy both model variants  and  0 to a real-world network of activities (i.e. a project 27,28 ), and explore two key quantities that characterise spreadingthe spreading event size (  ) and the rate by which spreading propagates (  ) -where  ∞ has an important effect.We subsequently focus on particular structural features of the pathways used to sustain these spreading events, and specifically on the different impact that  1 and  ∞ have.Finally, we explore the topological properties of nodes involved in large-scale spreading events.In agreement with recent studies 6,29 on information spreads, we report the presence of 'hidden influentials' (i.e.nodes with average topological properties which play a key role in sustaining large-scale spreading events), with their existence being increasingly pronounced at higher  1 and/or  ∞ values.

Results
We first establish the importance of indirect exposure (parameter  ∞ ) by illustrating its effect on the spreading event size through a comparative analysis between the results of spreading model  and  0 .We subsequently use model  to highlight the contrasting impact that direct (parameter  1 ) and indirect exposures have on propagation rate.We then focus on the relationships between the structural characteristics of these spreading events and their size/rate, and how  1 and  ∞ affect them.Finally, we provide insight on the topological features of nodes capable of fuelling large-scale spreading events, and how  1 and  ∞ influence them.

Spreading event size and propagation rate
We first define the ratio of the largest spreading event size obtained using the  model, over the largest spreading event size obtained under the  0 model, as . In a similar fashion, we denote the ratio of the average spreading event sizes as  avg .With  max and  avg being a function of both  1 and  ∞ , we can examine their isolated effect by considering their corresponding averages: . By applying the same approach to  avg , we obtain ̃a vg ( 1 ) and ̃a vg ( ∞ ).
Increasing parameter  ∞ leads to the values of  max (Fig. 2a) and  avg (Fig. 2b) being significantly higher than 1, demonstrating the augmenting effect that indirect exposures have on the spreading event size.This result suggests that this activity network contains a high enough number of non-trivial subgraphssuch as the ones included in Fig. 1 which allows  ∞ to have a significant impact on the progression of the spreading process.If the converse where to be true (i.e. max ≅  avg ≅ 1), it would suggest that the activity network could be approximated by 'locally tree-like' network, where indirect exposure would have had no effect over the spreading process, with results resembling that of a tree network (results from a tree network are shown in Supplementary Fig. 1 for reference).
Taken in isolation,  1 and  ∞ show qualitatively different characteristics in terms of their impact in the spreading process, further highlighting the non-trivial interaction between direct and indirect exposure in the context of a spreading process.The concave relationship between both ̃m ax and ̃a vg , with respect to  1 , demonstrates the principal role of  1 in sustaining the spreading process, with both ̃m ax and ̃a vg converging to 1 at the two extreme ends of  1 (Fig. 2c).On one hand, if  1 = 0 no spreading occurs and therefore the effect of  ∞ is nullified, with both  and  0 converging to identical spreading events; when  1 = 1 then direct exposure successfully switches the state of all nodes and therefore, no nodes are left for  ∞ to affect.Interestingly, high ̃m ax values are preserved up to relatively high  1 values ( 1 ≤ 0.7), indicating the strong influence of  ∞ even under unfavourable conditions i.e. under  1 = 0.7, a node is much more likely to switch state due to a direct rather than indirect exposure, and therefore one would naturally expect that the influence of  ∞ would be limited, giving rise to a low ̃m ax value.Finally, the intuitive expectation of  ∞ having an everincreasing effect in terms of both ̃m ax and ̃a vg is supported by the monotonically increasing trends shown in Fig. 2d.We now focus on the rate by which a given spreading event propagates across the network, which we quantify as   =    , where  refers to the average number of simulation steps needed for all nodes affected (within that spreading event) to switch state from 'non-affected' to 'affected' (which is a variation of survival probability 30 ).As such, we define the average propagation rate,  ̃ avg as the propagation rate for each spreading event, averaged across all events, and the maximum propagation rate,  ̃ max , as the propagation rate for the single largest spreading event.
We find that direct and indirect exposures have the converse effect with respect to the propagation rate, both in terms of  ̃ avg and  ̃ max .In particular, we find a positive relationship between  1 and the propagation rate, in terms of both  ̃ avg (blue marker) and  ̃ max (red marker), which corresponds to the intuitive expectation where increased direct exposure eases the way in which spreading progresses, enhancing the overall propagation rate (Fig. 3a).However, a negative relationship exists between  ∞ and propagation rate, both in terms of  ̃ avg and  ̃ max (Fig. 3b).This is due to the elaborate topology of the pathways deployed by the spreading process under higher  ∞ values, which delay the overall process (see section 'On spreading pathways').Considering both effects, this behaviour suggests that propagation rate is conflated by contrasting dynamics, where direct exposure provides immediateand thus, fasterpathways for spreading to propagate, while indirect exposure unlocks slower pathways which supress the overall in propagation rate (even though they may increase the overall spreading event size, as seen in Fig. 2d).

Topology of spreading pathways
We characterise the structure of a spreading event by considering the maximum depth and width of the underlying pathways that have sustained it.We define the maximum depth of a spreading event ( d ) as the shortest path between the initial seed node and the farthest node involved in the event 6 .In addition, we define its maximum width ( w ) as the maximum number of nodes affected whilst being at the same distance from the seed node 31 .As such, we characterise the structure of each spreading event as , which provides a continuous measure for the overall shape of the underlying pathways, where a high value of  / corresponds to long and narrow pathways, whilst a low value of  / corresponds to short and wide pathways.In that way,  / is maximised when the spread is composed of a single linear chain, and minimised when the spread resembles the structure of a star-shaped network.
We first focus on the relationship between the largest spreading event size,   max as a function of  / , where we identify a non-trivial relationship roughly composed of two opposing trends, see Fig. 4 (  max is normalised over the total number of nodes, ).The first trend dominates the small to medium sized events, where the spreading event size increases in step with  / , demonstrating the reliance of the spreading process to long and narrow pathways.However, as spreading events become larger than a given threshold (in this case, when and  / reverses, with the spreading process enlisting an increasingly high number of relatively shorter and wider pathways.This switch suggests the existence of an upper bound in the number of long and narrow sequence of consecutive tasks within the activity network.These sequences are largely composed of low-out degree nodes, and given the finite size of the network, pose a limit to the growth of the spread.To surpass this limit, and to further fuel the growth of the spreading event size, spreading utilises additional pathways which emerge through the inclusion of occasional high outdegree nodes, which allow for the spreading process to branch out in order to increase in size, resulting in relatively wider spreads. The rate by which the spreading event size increases depends on the spreading event structure.In the case where the spreading event size is negatively correlated with  / , the rate by which the spreading event size grows is roughly 3 times faster compared to the case where the spreading event size is positively correlated with  / (gradient is roughly -0.09 and +0.03, respectively).This result emphasizes the multiplicative effect that wider structures can provide, which in turn enhances the number of nodes that can be reached, and in turn, affected.
With respect to the impact of direct and indirect exposure,  1 and  ∞ show district trends, demonstrated by the marker colour patterns in Fig. 4a and 4b, respectively.Focusing on the impact of  1 , the transition in marker colour, from blue to red, is accompanied with a smooth increase in the spreading event size (Fig. 4a, 4c).This result is somewhat expected, since  1 plays a key role in the progression of the overall spreading process.In addition,  1 has an important role in determining the structure of the resulting spreading structure, albeit in a non-trivial manner.In particular, short and wide structures (low  / ) can occur at both extremes of  1 , with the shape slowly converging to the highest attainable  / values as  1 approaches ~0.5.Shifting focus to the impact of  ∞ , we observe that the entire range of  / is obtainable under any given value of  ∞ , indicating the limited role of indirect exposure in determining the structure of the pathways used by the spreading process (Fig. 4b).
The subtle impact of  ∞ on  / is further highlighted in the limited range of  / obtained reported Fig. 4d, which is significantly lower than the corresponding impact of  1 in Fig. 4c.
Note that these results are robust when we consider the relationship between   avg (instead of   max ) and  / , as a function of  1 or  ∞ , see Supplementary Fig. 2. We now focus on the relationship between the propagation rate of the largest spreading event,   max , and the structure of the pathways used to sustain it,  / , as a function of  1 (Fig. 5a) and  ∞ (Fig. 5b).Similar to Fig. 4, we identify a non-trivial relationship roughly composed of two distinct behaviours, where the propagation rate initially increases in step with spreading pathways becoming increasingly long and narrow (higher  / ).This overall increase in   max is the result of two conflicting effectsan increase in propagation rate, driven by higher  1 values (Fig. 5a and 5c), combined with a decrease in propagation rate driven by higher  ∞ values (Fig. 5b and 5d).This result demonstrates the conflicting nature of  1 and  ∞ , where the former relies on direct interactions which are faster to affect, while the latter introduces additional indirect pathways that take longer to evaluate completely due to their non-trivial nature (similar to the ones depicted in Fig. 1).
Once   max reaches a given threshold (in this case,   max ≈ 12), its positive relationship with  / , reverses to a negative relationship, which essentially reflects the need to utilise wider pathways (and hence, triggers a decrease in  / ), in order to increase the propagation rate further.This switch in behaviour is induced when  1 grows over ~0.5; as soon as this reversal in the relationship between   max and  / takes place, the fork-like shape of Fig. 5 indicates that two possible trajectories are available.Importantly, the similar marker colouring within both trajectories (Fig. 5a) suggests that  1 has a limited role in determining which of the two trajectories is followed.Yet the distinct marker colouring shown in Fig. 5b indicates that  ∞ is the key parameter in determining which of the two trajectories is followed.Specifically, the primary trajectory is accessible under the entire range of  ∞ , and the secondary trajectory, accessible under a limited range of  ∞ values, roughly ranging from 0 to 0.6.Both of these aspects are further reinforced by isolating the results obtained at equal  1 and  ∞ increments, with  ∞ controlling the emergence of the second trajectory (see Supplementary Fig. 3 for  1 , and Supplementary Fig. 4 for  ∞ ).

Fig. 5:
The relationship between the propagation rate of the largest spreading events across the entire range of  1 and  ∞ , (  max ), and its underlying structure ( / ), as a function of (a)  1 and (b)  ∞ ; (c) the relationship between the propagation rate of the largest spreading events, averaged over  ∞ and mapped as a function of  1 , and (d) the relationship between the propagation rate of the largest spreading events, averaged over  1 , and mapped as a function of  ∞ .
'Hidden influentials' and the effect of  1 and  ∞ Large-scale spreading events are typically associated with extra-ordinary topological characteristics of the node that initially triggers them (i.e. the seed node), the simplest of which being the (out) degree 32 our results confirm this common conjecture, albeit with certain strong caveats.Specifically, we find that the spreading event size is highly correlated with node out-degree, as reported in 6,33 (Fig. 6a).However, this correlation deteriorates as  1 increases in size, which suggests that the spreading process is shifting from being a local-driven process (and hence, dominated by the properties of the seed node) to a globally-driven process, where the characteristics of the intermediate nodes eventually dilute the correlation between spreading event size and the topological characteristics of the seed node.This correlation deteriorates faster once the effect of  ∞ is introduced, as additional intermediate nodes are employed early on during the spreading.Recent empirical work on information spreads has identified a similar effect, where large-scale spreading events are largely sustained by intermediate nodes with no special topological featuresthe so-called 'hidden influentials' 6,34 .
Following the work of Baños, et al. 6 , we evaluate whether these 'hidden influentials' exist in the activity network by comparing the average out-degree of nodes involved in a spreading event, excluding that of the seed node, ( ̃out avg ) with the network average ( out avg ), and the relationship between  ̃out avg and the spreading event size.Notably,  ̃out avg converges to  out avg as the spreading event size increases, confirming the presence of these 'hidden influentials' (Fig. 6b).This behaviour demonstrates that the existence of extra-ordinary nodes (e.g.hubs) is not a necessary condition for large-scale spreading to occur.More generally, this result highlights the intrinsic challenge in containing spreading in general, where system-wide spreading events are sustained by merely typical nodes, which themselves are hard to identify a priory.
We now focus on exploring the impact of  1 and  ∞ on the emergence of these 'hidden influentials', by considering the largest  ̃out avg , averaged across the entire  ∞ and  1 values, respectively.Notably, an overall decreasing trend is noted as both  1 and  ∞ increase, where an increase in  1 triggers a rapid decrease in  ̃out avg (Fig. 6c), while an increase in  ∞ triggers a linear decrease in  ̃out avg (Fig. 6d).This behaviour corresponds to an increase in the role of 'hidden influential' in sustaining the spreading process as immediate failure becomes more likely.In conjunction with the fact that spreading event sizes increases with larger  1 and/or  ∞ , these results further suggest that larger spreading events may be harder to contain than smaller ones, simply because larger ones are increasingly reliant on the existence of these 'hidden influentials'.

Discussion
In this paper, we have introduced a simple model which allows us to decouple the effect of indirect exposure from the overall spreading process, and comparatively examine its impact on key quantities, including spreading event size (Fig. 2) and propagation rate (Fig. 3).Our results highlight the conflating nature of spreading, where indirect exposure increases the number of nodes affected whilst reduces the rate in which the spread progresses.With indirect exposure being a derivative of clustered networks, our results clarify broader discussions within the literature, which typically focus in providing high-level insight [35][36][37][38] .For example, ref 36 focuses on identifying a (largely) positive link between clustering and spreading event size.Yet, the question of why clustering enhances spreading event size remains unexplored, with cases of no effect being treated as some sort of outliers.Our results suggest that indirect exposure is one possible avenue by which the relationship between clustering and enhanced spreading event size depends on, allowing for additional aspects to be explored in a similar fashion.
From a methodological standpoint, our results demonstrate the need to explicitly account for, and control the effects of, indirect exposure when modelling spreading-like processes.For example, consider the frequent use of 'locally tree-like' approximations, typically used to deploy analytically tractable expositions into various network dynamics 1,[39][40][41] .Despite the valuable insights that these approximations provide, the eventual nullification of indirect exposureand its effect on the spreading processclouds the real difference between these models and the respective real-world systems they represent, skewing our confidence and biasing results in a non-trivial manner.The results presented within this paper serve as additional motivation to recently emerging lines of inquiry 42,43 which focus on relaxing the 'locally-tree like assumption', integrating the effect of indirect exposure to the overall spreading process.
In the context of application, our work provides the grounds for a dialogue between researchers in the network science and project management, where hotly-researched, domain challenges (e.g.project complexity evaluation [44][45][46][47][48][49] ) can be treated as network-related problems 26,50 .For example, increased susceptibility to the spreading of failures can be reasonably interpreted as a contributing factor to project complexity.Hence, the relationship between spreading event size (or propagation rate) and the structure of the underlying pathways (Fig. 4 and 5 respectively) can serve as an objective, quantitative measure for project complexity.Similarly, the proportion of 'hidden influentials' within an activity network (Fig. 6) could be used to support the overall project risk mitigation scheme, where activities with limited connections (yet increased probability of being 'hidden influentials') receive adequate attention.

Data
The data comprises of a real-world engineering project which captures a set of planned activities that need to be completed in order to deliver a definitive commercial product in the area of defence.The overall duration of the project is 577 days, and is composed of 578 distinct tasks with 1,085 dependencies.Note that some tasks are used as planning instruments (e.g.milestones 51 ) and thus, include no dependenciesthese tasks are excluded from the analysis (8 tasks in total).
The delivery of each activity is typically conditional to a number of other activities.We refer to these interactions as functional dependencies, since they effectively control the function of each activity e.g. the start of activity  depends on the completion of activity .The directionality of each dependency dictates the functional role of each activity i.e. whether it acts as a predecessor (activity  proceeds activity ) or a successor (activity  succeeds activity ) to a subsequent task (leaf activities also exist, with no successor activities).
The set of tasks and dependencies was subsequently converted to an activity network, defined as a directed network  = {, }, where V is the set of nodes and E is that of directed edges.Every activity is abstracted as a node, where a functional dependency between activity  and  is captured in the form of a directed link from node  to node , denoted by   ∈ .The number of successors and predecessors each activity has corresponds to its out-degree and in-degree respectively.The cumulative probability distribution of out-degree (red) and in-degree (blue) is shown in Fig. 7 note its heavy-tail nature, evident by the straight line formed under the log-log axes.

Spreading Model Formulation
Every node  of the network at time  is characterised by a dynamic variable   () ∈ {0,1}, where '0' and '1' correspond to the 'non-affected' and 'affected' state, respectively.During the spreading process, node  may irreversibly switch from the 'non-affected' to the 'affected' state at time  if: (i) node  has at least one predecessor, node , and (ii) at least one node  was in the 'affected' state at  − 1.Then, we artificially switch the state of some seed node at  = 0, from 'non-affected' to 'affected' and track the progression of the spreading process as time increases at discrete increments of 1.
In order to distinguish between direct and indirect exposures, we keep track of node 's successors at time , say node , and assess whether node  has been encountered before.If so, then node  has been directly exposed to the failure of a different predecessor at some time > , but did not switch states during that time.Therefore, the probability of node  to switch states at now, at time , is controlled by  ∞ (Fig. 1, middle panel).However, if this is the first time node  has been encountered, then it is the first time node  is exposed to failure in general and therefore the probability to switch states at time  is controlled by  1 (e.g.Fig. 1, top panel).
Note the broad nature of the term 'affected', acknowledging the fact that failure can mean very different things, depending on the context of the project.For example, 'failure' can mean 'structural defect' in a construction project, or something much less tangible such as a 'contaminated' or 'compromised' in a cyber-security project.

Spreading Model Implementation
The model is implemented as follows.First, an initialisation phase is implemented, where simulation time  is set to 0 and the state of all nodes is set to '0'.In addition, an empty set  is created in order to record all successor nodes encountered during time .The spreading process is initiated by externally switching the state of node  from '0' to '1'.We then identify all successors of node , node(s)  (if no neighbours exist, the process terminates).For each node , we record index  in set (), and then check whether index  was already present in set ( − 1).If index  was not present, the interaction between node  and  is the result of direct exposure; hence, the probability of node  to switch states, under both model  and  0 , is equal to  1 .However, if index  was already present in set ( − 1), the interaction between node  and  is the result of indirect exposure; hence, the probability of node  to switch states, under model , is equal to  ∞ (in the case of  0 ,  ∞ is always set to 0).Once all node(s)  have been tested with respect to the prospect of changing states, we record the total number of state changes up to, and including, time ,   (), and increase  by 1.
The process repeats until the total number of state changes remains constant i.e.   () =   ( − 1).
Finally, the process is reiterated for each node , in order to evaluate the total number of state changes the failure of every possible seed node.Finally, this process is repeated for 48 independent runs, with results presented herein being the average.where variable Θ is uniformly drawn at random from U(0,1) for both eq.1 and eq.2.

Fig. 1 :
Fig. 1: Example to highlight the distinction between the direct and indirect exposure of node  to the failure of node  (time runs from left to right).Top panel focuses on the direct case, where node  is directly exposed to node 's failure; rest illustrate the case where node  is indirectly exposed, via node  (middle panel) and node  (bottom panel).

Fig. 2 :
Fig. 2: Difference in spreading event size between spreading model  0 and  (a) Ratio of the largest spreading event sizes ( max ) under the entire spectrum of  1 an  ∞ (largest  max = 3.2); (b) like (a), focusing on the ratio of average spreading event size ( avg ) under parameter  1 and  ∞ ; (c) ratio of largest (red) and average (blue) spreading event size, averaged across  ∞ , as a function of  1 ; (d) ratio of largest (red) and average (blue) spreading event size, averaged across  1 , as a function of  ∞ .

Fig. 3 :
Fig. 3: Propagation rate as a function of (a) direct exposure,  1 , and (b) indirect exposure,  ∞ .Blue and red markers correspond to the average rate across all spreading events,  ̃ avg , and propagation rate for the largest spreading event,  ̃ max , respectively.

Fig. 4 :
Fig. 4: Parameter space of the largest spreading event size (  max ), normalised over the total number of nodes (), and its underlying structure ( / ), as a function of (a)  1 and (b)  ∞ ; (c) the relationship between the largest spreading events, averaged over  ∞ and mapped as a function of  1 , and (d) the relationship between the largest spreading events, averaged over  1 and mapped as a function of  ∞ .

Fig. 6 :
Fig. 6: (a) Correlation coefficient between spreading event size and the out-degree of the initial seed node, as a function of  1 and  ∞ ; (b) average out-degree of nodes involved in a spreading event, excluding that of the initial seed node, ( ̃out avg ) as a function of the normalised spreading event size

Fig. 7 :
Fig. 7: Cumulative probability distribution for in-degree (blue) and out-degree (red) of the activity network.Note the heavy-tail nature of both distributions, evident by the straight plot line under loglog axes.