Modeling spatial social complex networks for dynamical processes
The study of social networks — where people are located, geographically, and how they might be connected to one another — is a current hot topic of interest, because of its immediate relevance to important applications, from devising efficient immunization techniques for the arrest of epidemics, to the design of better transportation and city planning paradigms, to the understanding of how rumors and opinions spread and take shape over time. We develop a spatial social complex network (SSCN) model that captures not only essential connectivity features of real-life social networks, including a heavy-tailed degree distribution and high clustering, but also the spatial location of individuals, reproducing Zipf’s law for the distribution of city populations as well as other observed hallmarks. We then simulate Milgram’s Small-World experiment on our SSCN model, obtaining good qualitative agreement with the known results and shedding light on the role played by various network attributes and the strategies used by the players in the game. This demonstrates the potential of the SSCN model for the simulation and study of the many social processes mentioned above, where both connectivity and geography play a role in the dynamics.
Much research has focused in recent years on a wide class of dynamical processes that take place in large human populations, at the scale of cities, whole countries, and even world-wide. Examples include epidemics spreading and strategies to arrest their spread Pastor-Satorras2001 (); Cohen2003 (); Belik2011 (), the evolution of the electoral map during elections Kim2003 (), the spreading of rumors Kwon2013 (), memes Hu2014 (); Gleeson2016 () and opinions Weng2012 (), the migration patterns of banknotes Brockmann2006 () and human populations Lee2014 (), and the effects of cities and infra-structure layouts on commerce and productivity Bettencourt2013 (); Schlapfer2014 (). Many of these questions require specific knowledge of individuals’ geographical location as well as their social contacts (many infections propagate by direct contact, or physical proximity; we discuss and influence the opinions of mostly those close to us, etc.).
In Milgram’s Small-World experiment milgram (), for example, participants were asked to pass a message (a postcard) to a person in a disclosed address, but only through a chain of social acquaintances: each participant was allowed to pass the message only to a person they know on a first-name basis. Of 160 messages started in Omaha, Nebraska, 44, or about reached the target in Boston, Massachusetts, with an average path length of about 5.4 links. How does the message find its way, let alone in such a short numbers of steps!?
Kleinberg’s seminal work kleinberg00 (), for nodes in a square lattice with random long-range connections, provided a first clue. This was later extended to fractal roberson06 () and anisotropic campuzano08 () lattices — still a far cry, however, from the geographical spread and network of connections typical of human society. Dodds, Muhamad and Watts conducted a large-scale online experiment that resembles Milgram’s original study, highlighting the role of information beyond just network structure dodds03 (). Liben-Nowell et. al. liben-nowell05 () proposed a spatial social network model with connections derived from an online bloggers community, and studied greedy routing on that model. Similar studies were conducted for online social networks scellato11 () and community structures from mobile phone records expert11 (); onnela11 () (see Ref. barthelemy11 (); barthelemy16 () for a more comprehensive review). Information on people’s location, along with their social contacts, is generally hard to come by and often relies on indirect proxies.
In Frasco2014 () we introduced a stochastic prototype Spatial Social Complex Network (SSCN), relying on just two controllable parameters, that simulates large populations, including the locations and the complex network of contacts between agents. This “baseline model” was designed with modest goals in mind: (a) The population density resembles the light density observed in satellite pictures of earth at night, (b) the population of “cities” (defined by percolation clusters Rozenfeld2008 ()) and their rankings follow Zipf’s law Zipf1949 (); Cristelli2012 (), (c) the social network of contacts exhibits a scale-free distribution, and (d) highly connected nodes tend to be located in denser and larger population areas. In addition to meeting these basic goals the SSCN baseline model also yielded good qualitative agreement with census data for the population density as a function of city size, and for the weak super-linear dependence of the cumulative degree of nodes in a city on its total population, as suggested from cell-phone data Schlapfer2014 (). Finally, it allowed us to shed some light on the weak deviations Rozenfeld2008 () from Gibrat’s law (that the rate of growth of a city and its fluctuations are proportional to its population size).
Despite these initial successes, the SSCN baseline model fails to mimic real SSCNs in some crucial ways: (i) The complex network of social contacts, while displaying a realistic scale-free degree distribution, is actually a tree, in contrast with the high degree of clustering observed in social nets and their proxies. (ii) The network of contacts is built through a redirection mechanism KR () which is an adequate description of how individuals might join a social network, but fails to account for the effect of relocations: every so often a person relocates to a far-away destination, for study, job, or other reasons. This creates particular correlations in the network of social contacts that are absent in the baseline SSCN.
In this paper we correct the baseline SSCN deficiencies with some simple adjustments. Connections to spatially closest neighbors are added to mimic the clustering effect in real social networks, and relocations turn out to be crucial in reducing the average path length between nodes. Simulations of Milgram’s Small-World experiment on the revised SSCN model achieve a good qualitative fit with the empirical findings. This demonstrates the suitability of the model as a substrate for simulations of other dynamic social processes that depend both on the contacts and the geographical locations of the agents.
Ii The baseline SSCN model
We now review the original, or “baseline” SSCN model, established in our previous work Frasco2014 (). The model produces a spatially embedded network where is a set of nodes and is a set of undirected edges. The spatial embedding of the network is encoded in the set of coordinates , where for 2D spatial networks (such as in our case) . A unique feature of the model is that it not only produces the requisite scale-free degree distribution for the edges but it also captures essential spatial features, such as a Zipf distribution of the populations emerging from the nodes clustering into “cities” Frasco2014 ().
Consider first the creation of nodes and edges in the baseline model, defining and . The starting point is an initial “seed” network, which in the baseline model consists typically of a single node. Nodes are added to the network one at a time, each contributing to a new edge, according to a variant of the Krapivsky-Redner (KR) model KR () with a single parameter — the redirection probability. Each time a new node joins the network, one of the existing nodes, , is chosen uniformly at random and is connected to directly with probability (creating a new edge ); otherwise, with probability , the connection is redirected to a randomly selected neighbor of (edge ). For large , this leads to a scale-free degree distribution kr_remark ()
Consider next the placement of the nodes in space, specifying . For a network of nodes, the baseline model places them within a square box of sides (with periodic boundary conditions). The initial seed node is placed at the origin, , and the location of subsequent nodes depends on whether it connects to node directly or to a neighbor , by redirection. If joins directly, it is placed at from (using polar coordinates), where the angle is chosen randomly from the uniform distribution, and is picked randomly from the distribution
where is the maximum possible distance between any two points within the bounding square. In the case of redirection, when node joins to , then we simply place at distance from at a random angle . The growth algorithm is illustrated in Fig. 1(a).
While the above growth rules were ultimately selected to best achieve the baseline model’s goals, they do make some intuitive sense as well. The redirection mechanism introduces a “rich-get-richer” bias in that redirection favors the random selection of nodes of a higher degree. This accounts for the emergence of the scale-free degree distribution. In addition, the connection and placement rules capture some basic ways of life: A person joins an existing social net when they are born. There is no choice in this matter and the social connection(s) established in this case is random (direct connection to node ). Eventually leaves home and settles at some distant location. The distribution of the distance to ’s new home, inversely proportional to the distance , is motivated by Kleinberg’s “magical” condition for navigability kleinberg00 (). The other possibility is that ’s most meaningful social connection happens through redirection ( is referred to a workplace or school, etc.) and in that case it makes sense to settle nearby to the new contact (at distance 1 — the minimal distance in our distance distribution).
The growth rules of the baseline SSCN model seem however too simplistic in that they account for a bare minimum of social connections: the connections to one’s birth place are represented by a single link, as are also the connections to people in a referred (redirected) situation. While the sparsity of connections can be justified on the grounds that the model is a scaled-down version of real life (fewer nodes, or people, so fewer contacts per person), there is no getting around the fact that the baseline model network of connections is a tree, in contrast with real-life social nets, where clustering is large (your friends have a higher than average probability to be friends among themselves). Another important effect is that of relocations: occasionally people move to a different place, sometimes more than once, over the course of their lives. When people relocate they maintain friendship with some acquaintances in their place of origin, and form friendships with their new neighbors. Thus relocations have a profound effect on the network of social contacts. In the next section, we describe a new version of the baseline SSCN that fixes these shortcomings.
Iii A revised SSCN model
For the present simulations we use a redirection probability , same as for the baseline model. This leads to a degree exponent which is typical of large-scale social networks BA2003 (); Newman2003 (). In addition to the significant changes that we made to the model’s connectivity, we made some minor changes to the boundary conditions and to the initial seed, and we describe these first.
Free boundary condition:
In the baseline model we used a bounding box of side and periodic boundary conditions. For the present work we adopt a boundary-free approach. Simply, the first node is placed at the origin and each subsequent node is placed in the same fashion as for the baseline model, but without regard to the bounding box. That is, the nodes are allowed to spread as far as the simulation takes them. Our simulations show that even with this free boundary condition the radius of gyration scales quite accurately as , so that the average population density per unit area remains constant even as the model is scaled up.
Starting with a single-node seed, as in the baseline model, tends to produce a few “megacities” — cities that are disproportionately larger than predicted by the Zipf distribution Zipf1949 (); Cristelli2012 (). In Frasco2014 () we showed how the problem might be overcome by starting with seeds consisting of several nodes. Here we employ a single-node seed, but let the redirection probability vary with the number of nodes added thereafter:
The probability converges rapidly to (we pick ), and the parameter controls the pace of the convergence. Thus, for the varying affects mainly the first nodes, but not the large-scale structure of the network. On the other hand, the fact that for the first few nodes reduces their capacity to attract further connections, thereby alleviating the problem of megacities. The effect of on the distribution of city sizes is shown in Fig. 2(a). In Fig. 2(b) we show the spatial layout of a typical network produced with , highlighting in color the first three largest cities. This very same configuration is used for the studies of connectivity and for the simulations of Milgram’s Small-World experiment reported below.
Closest neighbors and clustering:
We now come to the more serious revisions of the baseline SSCN model. A big issue is that the baseline model’s network of social contacts is a tree. This means that the probability for two of your friends to be friends amongst themselves is zero, while in real life that probability is in fact much higher than the average density of links, an effect best captured by the concept of clustering Watts98 (); Boccaletti2006 ().
To fix the problem of clustering in the baseline model, we now require that each node be connected to at least of its geographically closest neighbors, mimicking the fact that one indeed tends to befriend “next-door” neighbors. New edges are added in at the end of the growth process. The addition of new edges is illustrated in Fig. 1(b). Note that the baseline model corresponds to the special case of .
In Fig. 3 we plot the clustering coefficient of the network, , as a function of . We see that is quite large, and in line with real-life networks, already for . grows with (and decreases with the network size ) according to the empirical relation . The inset of the figure shows the dependence of the clustering coefficient of individual nodes upon their degree . The emergent relation () is also typical of many real-life networks Boccaletti2006 ().
The growth rules of the baseline model, even with the added rule for connecting closest neighbors, still fail to account for the very important effect of relocations. Every so often a person relocates to a new place, changing jobs or pursuing education, following marriage, etc. When a person relocates they retain many of their friendships at their place of origin, and form new friendships at their new location. This has a profound effect on the connectivity of the social network, as we shall see below. For now, however, we just describe the way to incorporate relocations in the revised SSCN model.
To relocate a single node we first pick two nodes and at random and move node to within distance from node , and at a random angle from , while retaining all of ’s connections. In the second stage, we examine the new environs of node and add the necessary connections to enforce the minimum closest neighbors rule. Note that the first stage entails merely changing , but not its contacts. The second stage ensures that agent not only keeps its old social connections, but also makes new acquaintances in the new place. The process of relocation is illustrated in Fig. 1(c).
The random choice of the relocating node and the target node (or location) is motivated by the “gravity model” for human mobility bharti08 (). It basically assumes that any individual is as likely to relocate as any other, and that relocating to any particular place (near ) is more probable the more populated that place is.
In the following section, we study the effect of migrating a fraction of the nodes in the system. A single relocation affects the degree of the relocating node in the same way as adding closest neighbors. (But note that undergoes two such updates.) Thus, the combined effect of connecting closest neighbors and migrating a fraction on the degree distribution is similar to that of connecting neighbors without migration. On the other hand, relocations have a dramatic effect on the pattern of connections and on navigation of the social network and they should not be neglected.
Iv Connectivity and Milgram’s Small-World Experiment
We now turn to the main question of how well the social network is connected and what we can learn from simulations of Milgram’s Small-World experiment. For concreteness, we study the typical SSCN configuration shown in Fig. 2(b) and focus on the connectivity between individuals in the largest and second-largest cities in the figure (population and , respectively). The two cities happen to be about units of length away from one another, which compares nicely with and with the actual span of the “country”.
iv.1 Shortest Paths
Consider first the shortest paths in the network. Shortest paths can be found very efficiently, for example by the Breadth-First Search (BFS) algorithm. The problem is that efficient algorithms such as the BFS require global knowledge of the whole network of contacts (or the full adjacency matrix). This type of information is clearly not available to any one person, so the mere existence of shortest paths cannot explain the results in Milgram’s Small-World experiment. Nevertheless, shortest paths constitute a useful “benchmark” to which one can compare various decentralized algorithms.
Since the SSCN network of social contacts consists of only one connected component (even in the baseline model) there exists a shortest path of links between any two nodes. We explore first how shortest paths evolve as one adds connectivity to the baseline model, first by connecting closest neighbors, then by migrating increasing fractions , , and of the nodes.
Our results for the shortest paths between nodes in City 1 and nodes in City 2 are summarized in Fig. 4. For the baseline model, the shortest paths between nodes in the two cities follow a bell-shaped distribution and average to just under 11 links. Adding connections to 5-closest neighbors reduces the shortest paths average length to about 8.5. This change is actually less impressive than one would expect: For a random -nodes network of average degree the average shortest path scales as . The baseline model has (it being a tree) and adding 5 closest neighbors increases to nearly 7. Thus the average looks long compared to the average expected for random nets, of . The reason is, of course, that the added links are not random, and — while important in accounting for the common phenomenon of “next-door” friends — they do not create efficient shortcuts. The situation is quite opposite for relocations: Migrating a mere fraction of the nodes results in an additional shortening of the average path lengths to about 7, a dramatic change for the tiny increase in , from to . Increasing the migration rate results in further reduction of the average path lengths, but the most dramatic change is that seen between no relocations at all and a tiny fraction of relocations. In that respect relocations seem to play a similar role to that of random long-range connections in the Watts and Strogatz Small-World networks Watts98 (). Finally, the insets in the figure show the distribution of path lengths for each successive change. The narrowing of these distributions can be traced to the homogeneization of the degree distribution as more links are added in.
iv.2 Greedy Paths
Consider now Milgram’s Small-World experiment milgram (). Participants in the experiment have access only to local information: You know who your friends are and where they live, etc., but have little information about their friends down the line. The puzzle is how the message finds its way, under these circumstances, let alone in a short number of steps. Local, or decentralized algorithms for passing the message may be quite involved and we shall test a few scenarios. For now however, we stick to the simplest greedy algorithm:
Pass the message to the contact that is geographically closest to the target (provided that it is closer than yourself).
Kleinberg kleinberg00 () had shown that, for his Small-World lattice, no other decentralized algorithm can obtain paths that scale more favorably with the population than the greedy algorithm. In other words, greedy paths give us a good idea of how well any other decentralized method might perform (at least functionally in ).
The proviso that each subsequent node is closer to the target is important: On the one hand it guarantees convergence; on the other hand, it means that the message might get stuck, when there is not a single contact that is closer to the target than oneself. In such a case there is no greedy path between the source and the target. When a greedy path exists, we say that the source and target are greedily connected. Greedy connectivity was explored for some benchmark networks (but not for SSCN models) in Sun2010 (). Some of the more important properties of greedy connectivity are:
Nodes that are connected in the usual sense might not be greedily connected (but not the other way around).
Greedy paths are never shorter than shortest paths.
Greedy connectivity is not transitive: If is greedily connected to and is greedily connected to , it is not necessarily the case that is greedily connected to .
Greedy connectivity is not symmetric: there might be a greedy path from to but no greedy path from to .
We have selected random pairs of nodes , with and , and then searched for greedy paths from to , and from to . The results are summarized in Fig. 5.
The average greedy path length for the baseline model, of about 7 links, is pleasingly short, however, only of the pairs are greedily connected. Adding connections to closest neighbors dramatically increases the greedy connectivity, to about of the pairs, but the average greedy path lengthens to about 39 links. These results can be understood as follows. In the baseline model the network of contacts is a tree and there is a unique path between any pair of nodes. (This path is also the shortest path.) Since the spatial connections are lain at a random angle , the probability that an -links path from to be also a greedy path, is . Thus the typical shortest paths, of average length , are greedy paths with probability , in general agreement with the observed result. Connecting closest neighbors makes for multiple paths between pairs of nodes. The probability that a greedy search might have to be abandoned at any particular step is roughly (assuming that the closest neighbors are randomly distributed, and neglecting the underlying baseline tree). For , the probability of the typical greedy paths (of length 39) making it through is therefore , quite in line with the observed results. Despite the dramatic increase in the success rate for greedy searches, the typical path length is too large to explain the observations in Milgram’s Small-World experiment.
Migrating even a small fraction of the nodes further increases the success rate, to about , but more importantly, it slashes the typical greedy path length by a factor of 2. Migrating larger fractions of the population achieves only modest improvements. Once again, the role of relocations seems analogous to that of random long-range connections in Watts and Strogatz Small-World networks Watts98 (). Nevertheless, the typical greedy path lengths, of about , even for migrations, still seems too long to account for Milgram’s results. Our SSCN model suggests that the difference is due largely to clever strategies adopted by participants in the experiment — people act more cleverly than the simple-minded greedy algorithm — and partly due to the effect of attrition: the finite probability to drop the search at any particular step selects for shorter paths. We turn to these issues next.
iv.3 Complex Strategies and Attrition
The greedy path algorithm cannot by itself explain the results from Milgram’s Small-World experiment and we are led to consider more complex strategies. A possible strategy is to prefer friends that live closer to the target to some extent, but give also some weight to friends that are exceptionally well-connected (since they might be more likely to make a better choice than ourselves). The following algorithm captures the gist of this idea.
Suppose that node currently holds the message that is destined for the (disclosed) target . Node assigns a score to each of his acquaintances ():
Here and are the geographical distances between and and and , and and are the degrees of node and of its -th contact, respectively. In other words, agent scores his acquaintances relative to himself (his own score is ), assigning higher value to friends that are closer to than himself, and that are better connected than himself. The parameter controls the relative importance of each attribute. With the scores at hand the strategy proceeds exactly as in the greedy algorithm, but with the aim of maximizing (rather than minimizing the distance):
Pass the message to the contact that has the largest score (provided that its score is larger than 1).
Kleinberg’s greedy algorithm corresponds to the special case of . For any other the strategy still guarantees convergence to the target (if a path is available), since the distance from to itself is zero, so that the score of is infinite and overwhelms all other considerations. (The case of is problematic, for the message may then fail to reach the target, even when is a contact of , and we therefore require .) The search for a path to is aborted when the proviso that is not fulfilled. In addition, for the path may revisit a previously touched node, creating a closed loop. The search is, of course, abandoned in such cases as well.
Fig. 6 summarizes the results of this mixed strategy, as applied to the case of closest neighbors and fraction of relocations. For clarity, we include only the results for searches from City 2 to City 1 (the small differences found for the reverse direction are discussed in the next subsection). Panel (a) shows the fraction of pairs, , that are successfully connected. The overall trend, shown in the inset, is of a rapid decay to zero as decreases. For close to 1, however, there is first an increase, from to a maximum of success rate for . At the same time, the average path length (Fig. 6(b)) decreases from at to at . There is in fact a whole range of for which the mixed strategy performs better (higher success rate and shorter paths) than the pure greedy algorithm of . At , for example, the success rate is as good as for , but the average path length is slashed by nearly 5 links.
As decreases beyond it becomes harder to judge the success of the mixed strategy: On the one hand there is the attractive effect of decreasing , on the other hand fewer and fewer pairs remain connected. One way out of this conundrum is to select the point for which matches the reported success rate of Milgram’s Small-World experiment, of roughly . This occurs for , where is reduced to nearly links.
An important conclusion is that geographical proximity is the largest factor in finding decentralized paths, as evident from the large values of that are optimal in our mixed strategy. This understanding is also in line with the findings of Liben-Nowell et al. liben-nowell05 (). Our mixed strategy shows that one can do better than geography alone (the case of ), yet not as well as reported by Milgram. The reason is that our mixed strategy fails to incorporate much of the intuition and social cleverness that are second-nature to people. In Milgram’s experiment, for example, the target’s occupation (stockbroker) was disclosed in addition to name and address. The name holds clues to the target’s gender and ethnicity, and the address might hint at social status. None of this information is accounted for in our naive approach.
A more realistic approach would probably still rely mostly on geography, at least until the message reaches the target’s city. Once inside the city the additional clues of occupation, gender, ethnicity, social status, etc., provide effective means for finding shorter paths (e.g., the stockbrokers in Boston tend to know one another). Indeed, subject reports in Milgram-like experiments strongly support this idea dodds03 (). The average path to the target’s city in our simulations is significantly shorter than the total path (Fig. 6(b)). At (where we reproduce Milgram’s success rate of ), for example, the average path length is , but only 4 of those links are needed to reach City 1. At this stage, Milgram’s results seem quite within reach.
So far we have considered attrition only due to the strategy, or strategical attrition: the search is dropped when the algorithm fails to find a next valid step. In real life, however, there are other reasons for defecting besides the unavailability of an attractive option: Participants may drop out from the experiment because of busyness, laziness, lack of motivation, etc. We refer to this effect as incidental attrition. We can lump both types of attrition into a single probability that an individual drops out of the experiment — this means a path of length has chance of being completed. From Milgram’s second study milgram69 (), for example, it can be estimated that . To illustrate the effect of incidental attrition, in Fig. 6(c) we plot the probability distribution for paths of length , , for the case of (solid line), along with the distribution that results from an incidental dropout probability of (broken line). As one would expect, the overall success rate drops, from to , but the (conditional) average path length is reduced by 4.3 links. The two types of attrition are a significant factor in the selection of shorter paths.
Consider finally the asymmetry of greedy, or decentralized paths: paths from in City 1 to in City 2 are not necessarily the same as paths from to . We see this effect quite clearly in Fig. 5, where the average path length for City is systematically shorter than for City , through all stages of the the model’s buildup. The success rates, too, are systematically smaller for paths from City 1 to 2 than the reverse (the differences are small and in the figure we put, for simplicity, only the average of the two rates).
A simple explanation to this asymmetry is that purely greedy paths from City 1 to City 2 can go through City 3, but those from City 2 to City 1 cannot (City 3 is farther away from the target), see Fig. 2(b). The situation is statistically symmetric for a “direct” commute, City , without City 3 in the picture: same expected number of successful paths and average path lengths in either direction. The extra routes tend to be longer than the direct commute, and account both for the higher success rate and the longer average path lengths in the City direction.
We observe small similar asymmetries also with our mixed strategy, for all values of . The region where the mixed strategy beats the pure greedy algorithm, for example, is somewhat narrower for the City direction, with (instead of for City ), but we do not have a simple explanation to account for these findings.
In summary, we have proposed improvements to the baseline SSCN model of Frasco2014 () that render it suitable for simulations of dynamic social processes, such as Milgram’s Small-World experiment milgram (); milgram69 (). The most important revisions call for connecting each node to a number of spatially closest nearest neighbors, to account for “next-door” friends, and relocating a fraction of the nodes, to account for relocations (due to job change, study, marriage, etc.). These two revisions have a minor effect on the degree distribution of the baseline model, but a dramatic effect on the connectivity properties of the network of social contacts: The connections to closest neighbors make for a robust clustering effect (absent in the baseline model), and even a tiny fraction of relocations introduces long-range connections that decrease the average path length between pairs of nodes substantially, similarly to the random long-range links in Watts and Strogatz’s Small-World networks Watts98 ().
Our simulations of the Milgram Small-World experiment show that Kleinberg’s greedy algorithm — based only on the geographical distance between nodes — is successful in finding decentralized paths between pairs of nodes, but the paths are too long to explain Milgram’s results. We have shown that more complex strategies, such as occasionally passing the message to acquaintances that are especially well-connected, can result in a significant reduction of the path length. We have also confirmed the notion that geography is the most important consideration in finding short paths dodds03 (); liben-nowell05 (), at least in the initial stages, until the message reaches the target’s city. The remaining path to the target, within the city, could be shortened considerably using the additional explicit information (e.g, occupation) and implicit information (ethnicity, social status) known about the target. We have also discussed the effect of attrition (the fact that participants drop out of the experiment for various reasons) and showed how it helps select for shorter paths.
Simulations of Milgram’s experiment pose a particularly strict test to the SSCN model, in that finding decentralized paths relies quite sensitively both on the location of the nodes and on their network of connections. The model’s success makes it a promising substrate for the simulation of other dynamical processes on social networks, where such considerations are important (epidemics, opinion models, etc.).
Acknowledgements.This work was funded in part by the Simons Foundation Grant No. 318812 and the Army Research Office Grant No. W911NF-16-1-0081.
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