Influence and Cascades December 10 2013 Group 34 • • The Impact of Influence on the Dynamics of Complex Cascades

Adriana Diakite,Emily Glassberg,Eyuel Tessema Stanford University [email protected], [email protected], [email protected]

I. Introduction current on the dynamics of complex information transmission by adding variation I. Motivation in influence across a navigable, small-world network model. We then explore the impact of ow do behaviors, contagions, products, variation in influence on real world informa- and spread through networks? tion cascades by comparing the results of our HThese questions can all be addressed simulated cascades to a cascade of retweets on using the of information the Twitter network. cascades. The impact of network characteris- tics on information cascades is of great interest to sociologists, politicians, public health profes- II. Prior Work sionals, marketing executives, and many others. Propagation through networks can be simple, Current research on complex propagation has meaning that contact with one "infected" node focused on random networks, regular lattices, is enough for transmission. Many diseases and small-world networks. A 2007 paper by spread in this fashion. However, propagation Centola et. al. investigated the impact of net- can also be complex, meaning that a certain work topology on the "critical threshold," the proportion of a node’s neighbors must have threshold above which no cascades occur. The adopted a behavior or before the node authors found that, in random networks, in- will decide to adopt it as well. Current liter- creasing the degree of each node decreases ature suggests that cascades involving social the critical threshold (promotes propagation). movements, attitudes, and ideas are likely to be However, in regular lattices, increasing the de- transmitted in this way (Granovetter 1978). As gree increases the critical threshold (inhibits a result, the dynamics of complex information propagation). Most interestingly, increasing cascades differ from simple cascades. Further- the randomness of regular lattices by rewiring more, the dynamics of social networks are in- edges also increased the critical threshold. Pre- fluenced by the specific relationships between vious literature has shown that increasing ran- individuals within that community. For exam- domness in the form of long-range links allows ple, it has been shown that people’s behavior in for increased transmission between neighbor- social networks is influenced by an individual’s hoods (Granovetter, 1973), however, the above status (Leskovec et al, 2010). Existing models results suggest that these conclusions do not of complex transmission are limited in that hold in the case of complex transmission. Ad- they treat the relationships between all pairs of ditionally, a 2010 paper by Centola found that nodes as equal, not accounting for potential ef- complex transmission occurred more rapidly fects of , interpersonal relationships, and with greater success (as determined by the and status on an individuals decision to adopt proportion of nodes that adopt the cascading a behavior or idea (Watts 2001, Centola et. al trait) in clustered lattices than in random net- 2007, Centola 2010). This project aims to extend works. The author suggests that this difference

1 Influence and Cascades December 10 2013 Group 34 • • results from increased social reinforcement due tance between the nodes. The distance be- to the high degree of clustering. These find- tween a pair of nodes will be the number ings support the claim that the dynamics of of steps to the most recent common ances- complex propagation are distinct from those of tor of that pair on the tree. This process simple transmission. Specifically, these results will be repeated until r edges have been demonstrate that complex and simple informa- added from each node to nodes in different tion cascades are impacted differently by net- neighborhoods (Model I). By varying the work topology. However, the network topolo- size of the neighborhoods (2h), and the inter- gies currently studied in the context of com- neighborhood connectivity (r), this general plex information cascades may be insufficient model will allow for the investigation of to capture the dynamics of cascades in real- variable influence on a variety of network world networks. For example, random graphs topologies. do not reproduce the clustering of real-world networks, regular lattices do not capture the b) Models of Variable Influence - Common small diameter seen in real-world situations, sense suggests that, in real world networks, and small-world networks are not navigable in distance between nodes is likely to affect the the way that real-world networks are. Further- influence that one node exerts on another. more, these analyses assume that each node For example, you are more likely to adopt in the network exerts equal influence on its the behavior of a member or close neighbors. A significant body of research on friend with whom you are often in contact the impact of properties of social groups such than the behavior of a distant acquaintance as status and homophily suggests that this as- that you rarely see. Therefore, we model sumption is highly unrealistic in real-world influence as a decreasing function of the settings. In this , we expand existing tree distance between a target node and its research on complex information cascades in neighbor (Model II). Specifically, the influ- two ways. First, we investigate the behavior ence of a node v on a node w is calculated of complex information cascades on navigable, as [ 1 ], where d(v, w) is the number of small-world networks. Second, we explore two d(v,w) steps to the most recent common ancestor models of variable influence and their impact of v and w on the tree. It is worth noting on the rate and success of complex information that Model I also indirectly incorporates a cascades. distance-based effect in terms of the abil- ity of one node to influence another (nodes II. Methods that are further away from a target node are less likely to have an edge to the target I. of Models and so are unlikely to influence it). For pur- poses of comparison, we implement a third a) Basic Model - The basic model we will use model wherein influence is modeled as an is an extension of the small-world network increasing function of the tree distance be- model (Watts and Strogatz, 1998). The net- tween a target node and its neighbor (Model work will be represented as a balanced bi- III). In this case, the influence of a node v nary tree of height H. The leaves of the tree on a node w is calculated as d(v, w). This are the nodes of the network. Nodes will be model could be interpreted as a scenario formed into completely connected "neigh- in which, as you become familiar with the borhoods" by placing an edge between ev- people around you, you become immune to ery pair of nodes in subtrees of height h. their views and no longer adopt behaviors Edges will be added between pairs of nodes from them. By contrast, when you meet in different neighborhoods with probabili- someone new, their behaviors or attitudes ties that decrease as a function of the dis- are surprising or interesting enough that

2 Influence and Cascades December 10 2013 Group 34 • •

you are more likely to adopt them. the graph adopt the new state), the adoption threshold is increased by 0.001, meaning that, relative to the previous simulation, nodes re- II. Description of Simulations quire at most one additional neighbor to have We generate a series of graphs each of H = 10, adopted a trait in order to adopt the trait them- varying h from 0 5 and r from 1 10. We then selves. The simulation is then repeated until we reach the adoption threshold at which cascades calculate the critical threshold (Tc) for each net- work topology (as specified by h and r)by fail. We say the of the critical threshold simulating complex cascades on each of these (Tc) is the value of the adoption threshold at the graphs. last successful cascade (0.001 below the thresh- old at the first failed cascade). We repeat this We simulate an in a procedure for each of the network topologies network by seeding a random neighborhood described above. with a new state. At each of one thousand Finally, we repeat this of experiments steps, each node will change state if a using each of the models of variable influence certain proportion of its neighbors had previ- described above. In these simulations, the con- ously adopted the new state. This proportion tributions of each neighbor towards the adop- is called the adoption threshold (discussed in tion threshold are weighted by the influence greater detail below). Each graph contains 210 score of that neighbor. (1024) nodes and, at every iteration, a cascade will either infect at least one additional node or the cascade will cease permanently. 1000 III. Analysis of simulations nodes is 98% of the 1024 nodes in the origi- ⇡ To compare the relationships between network nal graph, so, if a cascade has not succeeded by topology, critical threshold, and influence, we 1000 iterations, we know that the cascade must plot 3-dimensional graphs for each of our mod- have ceased prior to the final iteration. There- els with h, r, and Tc on the axes. This analysis, fore, performing more than 1000 iterations will however, only describes cascades in terms of have no impact on the final cascade size. success or failure. On a single network topology, we simu- To explore the behavior of cascades in late information cascades as described above greater detail, we created heatmaps to repre- at varying adoption thresholds. Because our sent the size - in terms of percent of nodes graphs are symmetric, every node in a network infected - of both the last successful cascade will have roughly the same degree. The degree (adoption threshold= Tc) and the first failed of each node (k) is approximately the num- cascade (threshold = Tc + 0.001) on each net- ber of neighbors within the node’s neighbor- work topology. hood plus the number of neighbors in different neighborhoods ((2h 1)+r). Given that the IV. Validation on real world data maximum h of our graphs is five, the maxi- mum r is ten, the maximum degree of nodes To test the relevance of our models to real in our network is 41, and the maximum dis- world cascades, we construct a three-tiered net- tance between two nodes is ten. Therefore, work using data from the Twitter network. We the minimum contribution of any one node start with a source user and retrieve two levels towards the adoption threshold is less than of neighbors. First, we retrieve the primary ( 1 ) ( 1 ) 0.002. Therefore, for the simula- neighbors; the followers of the source user. We 41 ⇤ 10 ⇡ tions on each graph, we initialize the adoption then retrieve the secondary neighbors; the fol- threshold to 0.001, meaning that nodes need lowers of the primary neighbors. Since each of at least one of their neighbors to have adopted our users has around 100 followers (this is the a trait in order to adopt it themselves. After average number of followers for Twitter users, each successful cascade ( > 90% of nodes in according to Lerman et. al, 2010), this should

3 Influence and Cascades December 10 2013 Group 34 • • leave us with a tree-like graph with 10, 000 these retweets for two . First, ’personal’ ⇡ nodes/users and 1, 000, 000 connections. tweets (personal here does not imply private) ⇡ A limitation of the current study is that may be less likely to reach users beyond the Twitter has harsh limits on the number of source user’s original friend group. Tweets queries one can make to an api endpoint (15 that did not originate with the source user are queries every 15 minutes). As a result, we have less likely to be personal, and therefore are not yet completed building the tree described more likely to be retweeted by her followers above. We hope to have it completed and ana- and be more interesting to analyze. Second, lyzed for the poster session on Friday. we assume that, in many cases the users in Once the graph is complete, we will simu- our graph retweeted each story because they late cascades using each of the three models encountered it through the source user. It is of variable influence described above. To ap- uncertain how valid this assumption is; the ply Model I, under which every node exerts source user and her followers could have dis- equal influence over every other node, we seed covered and chosen to retweet the tweet inde- the graph with a neighborhood comprised of pendently. Additionally, the source user could the source user and all of her followers. We be following one of her followers, and so the then simulate "cascades" at variable adoption tweet could have originated in a lower tier of thresholds and track how many secondary fol- the graph. However, it seems that we should lowers we expect to be infected. For Models not completely discount this number. II and III, where the influence a node exerts We will attempt to validate the methods on a target node is a function of the distance used in this analysis using real world infor- between those two nodes, the distance was de- mation cascades in the following ways. First, termined using the time zones of the users. we will compare the sizes of the simulated The distances between each pair of time zones cascades on our Twitter graph to the average are specified in Table 1 below. With these dis- retweet count of the source user’s tweets. This tances in hand and the influences of primary will allow us to assess the validity of our sim- followers over secondary followers weighted ulation procedure (if, on a real world graph, appropriately, the seeding and simulations are it can generate cascades similar to those seen performed as described for Model I. The pro- in the real world). Second, for each model of portion of users infected by each simulated variable influence, we will compare the criti- cascade will be recorded. Using the protocol cal threshold calculated on our Twitter graph described above, we will also use these simu- to that calculated on our model graphs. This lations to calculate a critical threshold (where will allow us to check the validity of our graph > 90% of nodes in the network are infected) for topologies (if, on the model graphs, cascades each of the three models of variable influence behave similarly to those seen on real world on the graph we created from the Twitter data. graphs). Third, what we would most like to To compare the results of this cascade to be able to do as a third step is to determine real world cascades originating from the source which of our models of variable influence most user, we extracted the source user’s 100 most closely represents how information cascades recent tweets. We also extracted the num- behave in the real world. To do this, we would ber of that each of these 100 tweets like to compare the critical threshold of the was retweeted counts for these tweets. We true retweet cascade to the simulated critical assume that the average number of retweets thresholds under each model of variable influ- across these tweets is the expected size of an ence. Unfortunately, it is impossible to deter- information cascade originating from this user. mine critical thresholds from data regarding Note that the 100 most recent tweets of the real world cascades. An information cascade source user include tweets that the source user either succeeds or fails; it is impossible to tell herself has retweeted. We chose to include under what conditions the same cascade would

4 Influence and Cascades December 10 2013 Group 34 • • behave differently. Another method would be ferent neighborhoods as a decreasing function to compare the sizes of real world cascades of distance. Thus, increasing the number of to that of our simulated cascades. This is lim- long-range links is likely to increase the de- ited by several factors. First, Models I and III gree of a node without increasing the number produce very similarly sized cascades (as does of infected neighbors it is connected to. This Model II except on very specific graph topolo- dilutes the impact of edges to nearby neigh- gies, when intermediate cascades can occur). bors that are infected, making cascades less As a result, it would be very difficult to distin- likely. However, inter-neighborhood edges are guish these models using this method. Second, essential for cascades to spread from one neigh- and perhaps even more seriously, in the real borhood to another. This suggests that there world, most cascades infect far fewer than 90%, will be some small or intermediate number or even 50%, of individuals in the network. of inter-neighborhood edges that will be suffi- Lerman et. al 2010 found that, in their sample cient to spread cascades, but small enough that of 140, 000 active Twitter users, the average the dilution of short-range edges will be lim- retweet count was 400. Furthermore, only 15 ited. At small neighborhood sizes (h <= 1), in- stories were retweeted more than 6, 000 times. creasing r inhibits cascades, causing Tc first to Therefore, even for tweets that clearly would be drop rapidly, then flatten off between Tc = 0.10 considered to have incited successful retweet and Tc = 0.05. At intermediate neighborhood cascades, the proportion of infected nodes in sizes (2 <= h <= 3), increasing r displays the network is only 0.04. This suggests that, no unidirectional effect with respect to Tc. At to understand information cascades on real large neighborhood sizes (h >= 4), increas- world networks, we need to rethink how we ing r slightly promotes cascades, causing Tc determine a successful cascade. to increase slightly, then flatten off around Tc = 0.125. So, when all nodes exert equal influence over a target node, increasing neigh- III. Results and Discussion borhood size appears to make cascades on the network more robust to increases in r. I. Tc as a function of h and r Model II - Similarly to Model I, in Model Model I - In general, the relationship between II, Tc is greatest when bot h h and r are low h, r, and Tc is saddle-shaped. Tc is the highest (h = 0, r = 2, Tc = 0.175). However, under this when neighborhoods are small and there is a model of influence, where nodes exert greater low degree of inter-neighborhood connectiv- influence over nodes that are closer to them, ity (h = 0, r = 1, T = 0.25). As high T the shape of the relationship between h, r, and c ⇡ c means that cascades are more likely to occur, Tc is much more plane-like than the saddle- these data suggest that, under the model of shaped curve seen in Model I. When inter- influence where all nodes exert equal influence neighborhood connectivity is small (r <= 3), on one another, cascades are most likely to oc- increasing h inhibits cascades (Tc decreases). cur when neighborhoods are small and there Furthermore, when neighborhood size is in- is limited connectivity between them. This termediate or large (h >= 2), increasing r finding is counterintuitive, but consistent with promotes cascades (Tc increases). This rela- the results found in Centola 2007. In complex tionship does not hold for small neighborhood information cascades, long-range links are un- size (h <= 1), in which case increasing r is likely to infect distant regions of the graph. not associated with a visible effect on Tc. In This is because nodes far away from the orig- general, cascades occur more readily under inally seeded neighborhood are less likely to Model I than under Model II. In Model II, be connected to multiple infected nodes. This nearby nodes are weighted more heavily than phenomenon is especially true in our model, more distant nodes. Thus, under this model, where edges are added between nodes in dif- the nodes that are weighted the most heavily

5 Influence and Cascades December 10 2013 Group 34 • • are the nodes within the neighborhood of the II. Proportion of nodes infected in target node. So, if a target node exists in an first failed cascade uninfected neighborhood, all of its most influ- ential neighbors are also uninfected. It will Under Models I and III, the first failed cascade therefore be more difficult to infect new neigh- (< 90% of nodes infected; adoption threshold borhoods under Model II than under Model I. = Tc + 0.01) infects between 0.1% and 6.25% However, when neighborhoods are small and of the nodes in the graph (Figure 4). This cor- inter-neighborhood connectivity is high, cas- responds to a single infected neighborhood cades occur more readily under Model II. Re- (the original seed), or, at most, two infected call that, under Model I, increasing neighbor- neighborhoods. By contrast, under Model II, hood size increases the robustness of cascades there exist graph topologies where up to 50% to increasing r. Under Model II, nodes ex- of nodes in the network become infected (e.g. ert greater influence over nodes close to them. h = 5, r >= 8, Figure 4). By definition, under This counteracts the effects of dilution; it in- all three models and at every graph topology, creases the robustness of cascades to increasing when the adoption threshold is at or below the r and promotes cascades, even on graphs with critical threshold, more than 90% of nodes in small neighborhoods. the network become infected. Model III - Under Model III, where nodes Cascades under Models I and III, therefore, that are farther apart exert more influence over display an "all or " behavior; either very one another than nodes that are close together, few nodes are infected or nearly the entire net- the relationship between h, r, and Tc is simi- work is infected. lar in shape to that seen in Model I. However, Under Model I, a source node exerts equal at small and intermediate neighborhood sizes influence over all of its neighbors. There are (h <= 3) and low inter-neighborhood connec- fewer edges between a source neighborhood tivity (r <= 3), when r is increased, cascades and neighborhoods that are farther away than are inhibited and Tc rapidly decreases. Under neighborhoods that are closer together, so it is Model I, increasing r at intermediate neigh- most likely that the seeded neighborhood will borhood sizes shows no clear, unidirectional first infect the neighborhood closest to it. This effect on Tc. This suggests that cascades under neighborhood also has few edges to neighbor- Model III are less robust against increases in r hoods at greater distances, however, with both than are those under Model I. Tc is frequently of the first two neighborhoods infected, there higher under Model III than Model I (cascades are up to twice as many randomly selected occur more easily). This can be understood inter-neighborhood edges. Infecting the sec- in the following way. Nodes within a neigh- ond neighborhood at distance 1 increases the borhood will, by definition, be closer together probability of infecting a further away neigh- than nodes that are in different neighborhoods. borhood. Similarly, infecting the third neigh- Consider a scenario in which a target node is borhood increases the likelihood of infecting located in a currently uninfected neighborhood. the fourth, and so on, hence the "all or noth- Many of the close neighbors of this target node ing" behavior. Under Model III, nodes further will also be within the uninfected neighbor- from the target node exert more influence than hood. However, it is possible that several far nearby nodes. Because the distance between away neighbors, in different neighborhoods, two nodes is determined by the number of will be infected. As a result, giving greater steps to the nearest common ancestor of the weight to far away neighbors increases the tar- pair, the nodes that exert the greatest influence get node’s chances of adopting the new state. over a target node are the nodes in the opposite This may explain the ease of cascades under half of the binary tree. Thus, if the adoption Model III relative to those under Models I and threshold is below the critical threshold, once II. a neighborhood is seeded, this neighborhood

6 Influence and Cascades December 10 2013 Group 34 • • is likely to infect every neighborhood in the IV. Work -Extensions of opposite half of the tree. The newly infected the Model half of the tree is equally far from the remain- ing neighborhoods of the first half of the tree The results of the simulations on our model (containing the seed neighborhood), and so graphs are promising and provide valuable in- those remaining neighborhoods are will be in- sight into how varying influence across nodes a fected as well. This leads to the observed "all network may impact how information spreads or nothing behavior." However, under Model through a network. However, much work re- II, cascades reach intermediate levels of suc- mains to be done in order to determine how cess, infecting only a portion of the network. variable influence operates to affect the dynam- The proportions of infected nodes in the first ics of complex information cascades on real failed cascade at each graph topology under world networks. We have considered several Model II are values of 1 , where i is an integer 2i variations of the model used in this analysis i between 0 and 10. This is because there are 2 that may improve its relationship to real world nodes at distance i from the source node. Un- data. Future analyses could benefit from ad- der Model II, cascades are very likely to infect dressing each of the extensions discussed be- i these 2 nodes at small values of i. However, low. First, our current model sets adoption when i is small, there are few nearby neigh- thresholds to be the proportion of neighbors borhoods to infect (1 neighborhood at distance infected. It would also be possible to set the 1, 2 at distance 2, 4 at distance 3, etc.). Once adoption thresholds to be the absolute num- that small number of nodes in nearby neigh- ber of neighbors infected. This would remove borhoods has been infected, for the cascade the dilution of edges to infected neighbors that to spread it must overcome a greater hurdle; we see in the current model with increasing r; i it must infect the 2 nodes at the next highest nodes would no longer be penalized for having value of i. Thus, as more neighborhoods be- large numbers of neighbors. At this point, it is come infected, it will become more difficult to unclear which choice is a better representation infect the remaining neighborhoods. Therefore, of real world information cascades. Second, as the adoption threshold increases, first cas- we could implement a fourth model of variable cades fail to infect the farthest part of the graph influence. In the models of influence discussed (the opposite half), then they fail to infect the in this analysis, the sole factor in determining next farthest part, etc. the influence one node exerts on another is the distance between the influencing node and its target. However, it is also possible that the influence a node is able to exert on its neigh- III. Validation - Cascades of Retweets bors is, at least partially, determined by the characteristics of that node. For example, if a on Twitter person is particularly charismatic or particu- larly well-respected in his or her field, he or As previously mentioned, data regarding she may be more likely to influence all of his retweet cascades on Twitter are forthcoming. or her neighbors, regardless of how closely the The following are some basic statistics regard- two people are associated. The impact of this ing the tweets we collected. Note the very type of status on complex information cascades small average retweet count of tweets originat- could be explored by randomly assigning an ing from the seed user and the very large aver- influence score to each node in the graph (See age retweet count of tweets that were retweeted Figure 5, Model IV). In simulations, all edges by the user. This justifies our decision to in- originating from this node would be weighted clude the source user’s retweeted tweets in our by this influence score. It is not yet known analysis. whether individual characteristics, such as sta-

7 Influence and Cascades December 10 2013 Group 34 • • tus within a community, or relative character- References istics, such as distance between individuals, a more significant role in real world in- [1] Centola, D. (2010) The spread of behav- formation cascades. It seems likely that both ior in an online . Science will be relevant in predicting how ideas will 329:1194-7. spread through networks. It is also unclear [2] Centola, D, Macy, MW, Eguiluz, VM. how best to calculate "distance" between in- (2007). Cascade dynamics of multiplex dividuals to predict how they will influence propagation. Physica A 374: 449-56. one another; geographic distance, number of shared communities (e.g. did two people go to [2] Granovetter, M. (1973). The strength of the same high school, work at the same com- weak ties. Am J Sociol 78: 1360-1380 pany, etc), number of shared interests (e.g. did two people comment on the same blog post or [3] Granovetter, M. (1978). Threshold mod- read the same journal article). To understand els of behavior. Am J Sociol 83: how our models of variable influence relate 1420-42.0 to real world networks, greater exploration of [3] Kempe, D, Kleinberg, J, Tardos, E. various distance metrics on real world cascade (2003). Maximizing the spread of influ- data is required. Third, in this analysis, we ence through a social network. SIGKDD. assume that each node in the graph has the same adoption threshold. This assumption [3] Romero, DM, Meeder, B, Kleinberg, J. is unlikely to bear out in real-world networks. (2011) Differences in the mechanics of in- Variable adoption thresholds is likely to impact formation diffusion across topics: idioms, cascade dynamics and is therefore worth fur- political hashtags, and complex contagion ther analysis. Fourth, the graphs generated in on Twitter. Proc WWW. this analysis are symmetric. In real-world net- works, however, the degree distributions follow [3] Leskovec, J, Huttenlocher, D, Kleinberg, J. power-. This means that there are likely (2010) Differences in the mechanics of in- to be some very highly connected individu- formation diffusion across topics: idioms, als or neighborhoods and many smaller, more political hashtags, and complex contagion sparsely connected groups. Again, this alter- on Twitter. Proc WWW. ation in network connectivity is likely to affect [3] Lerman, K and Ghosh, R. (2010) Infor- the dynamics of multiplex cascades. For exam- mation contagion: An empirical study of ple, if a cascade infects a highly connected indi- the spread of news on Digg and Twitter vidual or neighborhoods, it may very quickly Social Networks. spread to a large portion of the graph. To cap- ture this, we could vary connectivity across [3] Watts, DJ. (2001) A simple model of global neighborhoods (e.g. vary h from one neighbor- cascades on random networks. PNAS 99: hood to another, Figure 5, Model V). 5766-71. [3] atts, DJ and Strogatz, SH (1998). Collec- tive dynamics of âAŸsmall-world⢠A˘ Z´ net- works. 393: 440-2

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Figure 1 – Representation of models. Here networks are shown with tree height H=3, neighborhood height h=1, number of inter-neighborhood random edges r=1. The darker a node, the higher its status. In Model II, the target node is denoted by an asterisk.

Figure 2 – Flowchart representing calculation of critical thresholds. Start with a specific graph topology [h initialized to 0, r initialized to 1]. Adoption threshold initialized to 0.001. Run simulation (start at star). When updating r, r=r+1. When updating h, h=h+1. When updating adoption threshold (Ta), Ta=Ta+0.001.

Figure 3 – Plots showing critical thresholds as a function of h and r for each of the three models of variable influence, and for each of the pairwise comparisons between the models.

Figure 4 – Heatmaps showing the cascade ratio - in terms of percent of nodes infected - of both the last successful cascade and the first failed cascade on each network topology.

Figure 5 – Representation of potential future models. Here networks are shown with tree height H=3 and number of inter-neighborhood random edges r=1. In Model IV (top), status is randomly distributed across nodes. The darker a node, the higher its status. In Model V, neighborhood height h=0 for the leftmost neighborhood, 1 in the middle, and 2 on the right. In this representation, all nodes have equal status, but this model could be combined with any of the models of status described in this analysis.