Multiplexing in Networks and Diffusion

We study two different data sets of multiplex networks in a total of 143 villages, both from the state of Karnataka, India, covering a total population of nearly 30,000 households.

We begin with some descriptive statistics on the network layers. A link is present in a given layer if either household named the other household in one of the questions in that category (e.g., we code a kerorice link if either household reports borrowing kerosene or rice from or lending it to the other household). In terms of notation, we define a multi-layered, undirected network for each village $v$,⁵⁵5One can also define directed networks from our data, which we comment on at several points below. Directed links open some important but tangential questions for multiplexing, which we leave for further research. for layer $\ell=1,\ldots,L$, with $g_{ij,v}^{\ell}=1$ if either household $i$ or $j$ reported having a relationship of type $\ell$. We add another layer where $i$ and $j$ are linked if they belong to the same jati. For the Microfinance Village Sample, where GPS data are available, we construct a weighted graph where the $ij$ entry is the geographic distance between the two households.

The union layer has a link present if a link exists in any layer. The intersection layer has a link present if it exists in all layers.⁶⁶6Both of these definitions include the jati layer but exclude geography, since we are able to define the geographic layer for only one of our datasets, and it is a weighted network in any case. We make these definitions to maintain consistency of the meaning of the union and intersection layers across the two data sets.

We also build a weighted and directed network whose edge weights are the sums of indicators for links in all directed layers (thus excluding jati and geography). We call this the total network. Finally, below we describe another weighted and directed aggregate network that we build from the principal component analysis.

The empirical analysis demonstrates that multiplexed networks in our data are rich and embed important information that would be lost by collapsing them into a single summary measure. A natural next question is how this distinction between layers matters for outcomes of interest. Here we focus on diffusion in the RCT villages.

We proceed as follows. Based on prior work, we would expect that more central seeds should lead to greater diffusion (Banerjee et al., 2013, 2019). Those papers defined a single network by using the union network and computed diffusion centrality based on that. However, given our rich multiplex data, we note that the seeds’ diffusion centrality differs across layers. Thus, we examine which layer is the most predictive of diffusion in the RCT if we were to compute diffusion centrality based on that layer alone.

We use a specific diffusion centrality measure developed in Banerjee et al. (2013) and further studied in Banerjee et al. (2019). In particular, the diffusion centrality of a node $j$ in layer $\ell$ in village $v$, $DC_{j,v}^{\ell}$ is defined by

where $T$ is the number of rounds of communication and $q$ is the probability of transmission in each period across any given link. Following Banerjee et al. (2019), for village $v$ and network layer $\ell$, we set $T=\operatorname{diameter}(g_{v}^{\ell})$, and $q=1/\lambda_{v}^{\ell}$, where $\lambda_{v}^{\ell}$ is the largest eigenvalue associated with $g_{v}^{\ell}$.¹²¹²12See Banerjee et al. (2019) for a theoretical foundation for using these as default settings in diffusion centrality. We calculate the diffusion centrality of the seed set of village $v$, $S_{v}$, for layer $\ell$ by

We then can calculate how diffusion varies with the diffusion centrality of the randomly assigned seed set under layer $\ell$ by regressing

where $y_{v}$ represents the number of calls received from village $v$ (a measure of diffusion of information), and $X_{v}$ includes controls for number of households, its second and third powers, and number of seeds assigned in that village. We standardize all the regressors.

Table 2 depicts how differently the layers predict diffusion based on our specification in (2.1). (Supplementary Appendix Figure S1 plots the $90$% and $95$% confidence intervals.)

The advice layer stands out as the most predictive, and we see that the kerorice and social layers are also significantly predictive. Notably, consistent with what we observed in the correlations and principal component analysis, jati explains the least of the variation and is not significant.

Interestingly, the four synthetic networks we have mentioned that aggregate the layers in specific ways—union, intersection, total, and backbone—all perform worse than the individual layers with the exception of jati. However, this appears to be rooted in the inclusion of jati in those aggregates. In Supplement A.2 we recreate Table 2 with aggregate layers that omit jati in their construction (this applies only to union, intersection, and backbone). This improves their performance, with the backbone network now yielding an $R^{2}$ second only to the advice layer.

Given how correlated the layers are, we also perform a LASSO ($\ell_{1}$-penalized) regression to select a sparse set of relevant variables that explain diffusion. We then use post-LASSO least squares to estimate how seed set centrality under the selected layer(s) affects diffusion.

The regression of interest is given by

where the variables are as described in (2.1) and instead of running a separate regression for each layer, we now include all the layer variables simultaneously. We are interested in which $\beta^{\ell}$ are estimated to be non-zero and the consistent estimates of these parameters.

A complication we face here is that in order to be consistent, LASSO requires a condition called irrepresentability, which requires the regressors of interest not to be excessively correlated (Zhao and Yu, 2006). In our setting, this requirement fails since the network layers are highly correlated. To overcome this problem, we use the Puffer transformation developed by Rohe (2015) and Jia et al. (2015), which recovers irrepresentability when the number of observations exceeds the number of variables. Although the regressors $(DC^{\ell}_{v})_{v,\ell}$ have correlated columns, by appropriately pre-conditioning the data matrix, we can force its columns to be orthogonal and therefore irrepresentable. Puffer-LASSO then recovers the set of relevant variables with probability tending to one exponentially fast in the number of observations, with consistent parameter estimates, that are asymptotically normally distributed with probability approaching one (Javanmard and Montanari, 2013; Jia et al., 2015; Taylor and Tibshirani, 2015; Lee et al., 2016; Banerjee et al., 2024a).

We see the results in Figure 4, where we plot which layers are selected by the LASSO as we increase the penalty level, forcing LASSO to select fewer variables. We find that, at the highest penalty level, only the advice network layer is selected, with the post-puffer LASSO OLS regression in Table 3 depicting a 64% increase in diffusion relative to the mean ($p=0.011$). Despite the fact that multiple layers are useful in explaining diffusion, neither the backbone, the union, nor the intersection network proved to be the most useful.

The fact that centrality in the advice layer is singled out as the best predictor of diffusion under sufficiently high penalty does not mean that the other layers have no impact on diffusion. In fact, a combination of the layers still provides significantly more prediction than just the advice layer, as shown in Table 4.

Table 4 presents both cumulative and marginal F-tests as variables are added in the order selected by LASSO. We can see that adding intersection is marginally significant above advice, and further including kerorice and jati yields a more complete model, with an improvement significant at the 5 percent level.¹³¹³13 In Appendix Table S4 we exclude the extra layers of intersection, union, and backbone, which are “constructed” layers that are derived from these basic layers. F-tests include the basic layers in the order selected by the Lasso. Thus, even though jati serves as a poor substitute for other layers, it turns out to be a useful complement to them in predicting diffusion.

Next, we examine how diffusion depends on the extent to which the layers in a village are multiplexed. Specifically, do villages with greater correlation among their network layers experience higher or lower levels of diffusion? To do this, we first develop a measure of the extent to which a village is multiplexed.

We begin by defining a multiplexing score for household $i$ in village $v$ as

The multiplexing score for a household $i$ measures the average fraction of relationship types it has with each of its neighbors. The numerator calculates the average number of links household $i$ has to each neighbor across all $L$ relationship types. It does this by first summing the number of links between household $i$ and each neighbor $j$ across all layers, dividing by the total number of layers $L$, and then summing this average across all neighbors $j$. The denominator counts the number of unique neighbors of household $i$ by summing an indicator for whether there is at least one link between $i$ and $j$ across any layer. For example, $m_{i,v}=1$ if whenever household $i$ has a relationship with some other household $j$, then it has all possible relationships with that other household. In contrast, when there is no multiplexing, this measure would be $1/L$.

We aggregate this to the village level by taking $m_{v}:=\frac{1}{n_{v}}\sum_{i}m_{i,v}$. Further, we define a dummy variable for having an above-median amount of multiplexing in the sample as

Our regression of interest is

where $DC_{v}^{advice}$ denotes the diffusion centrality of the seed set in village $v$ for the “advice” layer (which was singled out as the best predictor of diffusion).

Here, $\zeta$ captures the returns to increasing the diffusion centrality of the seed set. Since information is seeded in all networks, $\eta$ captures how the extent of diffusion changes with the worst possible seeding (the theoretical intercept). The coefficient $\beta$ captures how incrementally improving seeding differentially affects the extent of diffusion as a function of multiplexing.

The interaction term $DC_{v}^{advice}\times\text{High Mpx}_{v}$ is particularly important, and its coefficient of primary interest, since villages with low seed set centrality experience very little diffusion, and hence multiplexing has a very limited opportunity to make any difference in diffusion. Thus, multiplexing’s marginal impact (positive or negative) should be most pronounced in settings where the seed set centrality is high.

Table 5 reports the coefficient estimates. As expected, the coefficient on seed set centrality is positive and significant. We also find that both $\beta<0$ and $\eta<0$. Qualitatively, $\eta<0$ indicates that more multiplexed networks generate less diffusion, with the caveat that these villages could be different for other reasons, and the coefficient is not significant. Importantly, the coefficient $\beta<0$ indicates that villages with more central seeding—and thus higher levels of diffusion—have their diffusion impeded by multiplexing.

We study diffusion/contagion in a society consisting of a finite set of individuals $N=\{1,\ldots,n\}$. Each individual has relationships captured via layers $\{1,\ldots,L\}$, with a generic layer represented by $\ell$. In each layer $\ell$, the interactions between individuals are described by a (possibly directed) network with adjacency matrix $g^{\ell}\in\{0,1\}^{n\times n}$, such that $g^{\ell}_{ij}=1$ if there is a link from $i$ to $j$ in layer $\ell$ (interpreted as being capable of being infected by $j$, e.g., via $i$ paying attention to $j$ in a model of information flow), and $0$ otherwise. We denote the multigraph consisting of $L$ layers by $g=(g^{1},g^{2},\ldots,g^{L})$.

Let $\mathcal{L}_{ij}=\{\ell\mid g^{\ell}_{ij}=1\}$ denote the set of layers in which there is a directed link from $i$ to $j$. The set of all neighbors for a given node $i$ is denoted $\mathcal{N}_{i}=\{j\mid\mathcal{L}_{ij}\neq\emptyset\}$.

To track infection across time, we index discrete periods by $t\in\{0,1,2,\ldots\}$. At each point in time, an individual in the network is in one of two states: Susceptible (S) or Infected (I). The status of individual $i$ at time $t$ is denoted by the random variable $x_{i}(t)$. If $x_{i}(t)=1$, individual $i$ is infected at time $t$; if $x_{i}(t)=0$, individual $i$ is susceptible at time $t$. The state of the society at time $t$ is given by the vector $x(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))\in\{0,1\}^{n}$.

At each time $t$, an individual’s state can change based on the infection status of its neighbors. A susceptible individual $i$ becomes infected if it receives at least $\tau$ infection transmissions from its infected neighbors in a given time period. An infected individual recovers (and becomes susceptible again) randomly with a probability $\delta$ at the end of a period. If $\tau=1$, this represents a standard (simple) contagion process, while with a threshold $\tau>1$ this is known as a complex contagion (Granovetter, 1978; Centola, 2010).¹⁴¹⁴14This is closely related to games on networks (Morris, 2000; Jackson and Zenou, 2014).

To complete the description of the model, we examine the mechanics of contagion in more detail. Given that individuals can be connected via multiple layers, we need to define how transmission occurs through multiple layers. Let $x_{ij}(t)$ represent the (random) number of infection transmissions at time $t$ to a susceptible node $i$ from an infected node $j$, conditional on $j$ being infected.¹⁵¹⁵15This is related to the modeling of dosed exposures in the literature on contagion; see Dodds and Watts (2004). At most one transmission can take place per layer. We denote the distribution of infection transmissions from node $j$ given $\mathcal{L}_{ij}$ by

This is the probability of $k$ transmissions; note $f(k;\mathcal{L}_{ij})$ can capture arbitrary patterns of correlation in infection transmission through multiple layers. For each layer $\ell$, let $q_{\ell}\in(0,1)$ be the marginal probability of infection transmission from an infected individual to a susceptible one if they are connected via that layer. We allow different layers to have different contact probabilities, which is needed given the heterogeneity in the roles of different layers discussed in Section 2.3. If there is a positive correlation in transmission across layers, two nodes connected by layers $\{A,B\}$ have an infection distribution satisfying $f(2,\{A,B\})\geq q_{A}q_{B}$.

The probability that a susceptible individual $i$ becomes infected at time $t$ given the infection status of its neighbors at time $t-1$ is

We first analyze the case of simple contagion, $\tau=1$. We focus on the case of two layers as this captures all of the essential intuition.

The results on simple contagion are unambiguous: multiplexing impedes simple diffusion/contagion except in extreme cases of negatively correlated transmission probabilities. Complex contagion, in contrast, presents a more nuanced picture. Multiplexing can both enhance and impede diffusion, depending on the circumstances.

In complex diffusion, two competing forces of multiplexing emerge. One force mirrors the effect seen in simple contagion: diversifying links increases the probability of at least some links reaching infected individuals. However, a counterforce now exists: conditional on reaching an infected individual, multiplexing leads to higher probabilities of multiple transmissions, compared to spreading those links across other individuals who might be uninfected. This makes it more likely that a contagion threshold greater than $1$ is reached.

To keep the analysis as uncluttered as possible, we again focus on the case of two layers. We also consider a case where the correlation in transmission across layers is neither too high nor too low, so that there is an $\varepsilon$, to be determined in the proofs of the propositions below, for which $(1+\varepsilon)q_{A}q_{B}\geq f(2,\{A,B\})\geq q_{A}q_{B}$. Of course, a sufficient condition for this to hold is independent transmission. This condition is needed as with excessive positive or negative correlation in transmission, strange discrete behavior in transmission as a function of multiplexing can occur.¹⁹¹⁹19For instance, if transmission is perfectly positively correlated, then one is always more likely to get two transmissions from a single multiplexed connection than two unmultiplexed connections, but is always more likely to get one transmission from the reverse. This then implies that the optimal configuration of connections depends on whether $\tau$ is even or odd, in complicated ways as a function of a node’s overall degrees in each layer. The restriction to two layers allows the results to highlight the more fundamental forces of multiplexing.

The intuition behind this result is as follows. There exist nodes $i,j,k$ such that under $g$, node $i$ is connected to $j$ on two layers and to $k$ on none, while under $\widehat{g}$, node $i$ is connected to $j$ on one layer and to $k$ on the other. The cases in which this difference can be pivotal are when the other connections to other nodes have led to either $\tau-1$ or $\tau-2$ transmissions. With high infection and transmission rates among neighbors, the $\tau-1$ case predominates, making the situation resemble simple contagion—thus, less multiplexing leads to a higher chance of infection. Under low infection rates, the $\tau-2$ case becomes more likely, requiring two incremental infections. This is highly improbable across two separate neighbors but more likely with a single neighbor, making more multiplexing advantageous for infection probability.

Interestingly, as we will see in the simulations below, these forces can interact non-monotonically in the intermediate range for infection and transmission rates, which explains the gap between the upper and lower bounds.

We now state how this translates into an aggregate infection rate.

Note that the steady-state infection of every connection type shares the same ordering as the overall infection rate.

The theoretical results are based on the asymptotic statistical behavior of large random networks. To see how the results work in smaller empirical networks, we run simulations on the networks from the RCT villages. We simulate a Susceptible-Infected-Susceptible (SIS) diffusion process for the cases of both simple and complex diffusion and compare outcomes as multiplexing is varied. In order to compare across similar-sized networks where only multiplexing is changing, we take a given village network and construct two-layer networks by combining different pairs of empirical networks (which end up empirically having different multiplexing rates), in a way we specify below. We then perform many diffusion simulations on these two-layer networks for each village.

More specifically, for each village, we begin by picking three empirical adjacency matrices representing different network layers sorted in decreasing order of their average out-degree: $A_{1}$, $A_{2}$, and $A_{3}$. We then pair $A_{1}$ with $A_{2}$ for one simulated diffusion, and $A_{1}$ with $A_{3}$ for the other. To ensure that the average out-degree is comparable across the networks, we prune at random the links in $A_{2}$ to match the average out-degree of $A_{3}$, resulting in a pruned network $A_{2}^{{}^{\prime}}$. We construct two multiplexed networks: $g^{\prime}$, by combining $A_{1}$ and $A_{2}^{{}^{\prime}}$, and $g$, by combining $A_{1}$ with the $A_{3}$. The process is presented in full detail in the Appendix (Algorithm 2).

The diffusion process (also presented in full detail in the appendix in Algorithm 1), is as follows. First, a susceptible node can get message transmissions from each infected neighbor in each layer, i.i.d., with probability $q$ in each period. Second, a susceptible node gets infected only if it receives at least $\tau\geq 1$ contacts in a given time period and the count resets in each time period. Third, in each period an infected node transitions back to being susceptible with probability $\delta$. We terminate the simulation when the share of infected nodes changes by less than a small threshold between consecutive iterations. In our simulations, we use $\tau=1$ for simple diffusion and $\tau=2$ for complex. In each simulation we set the number of randomly selected seeds in the initial period to be $\lfloor\sqrt{n}\rfloor$, where $n$ is the number of households in the network. For both, simple diffusion ($\tau=1$) as well as complex diffusion ($\tau=2$) we run simulations on a grid of $(q,\delta)\in[0.1,0.5]\times[0.1,0.5]$. We run the diffusion simulations $500$ times for each village across both multiplexed networks $g,g^{\prime}$ described above. We report the averages across all $70$ villages.

Given that these are smaller networks, some simulations end up randomly having more or less diffusion in any given run across the two comparison networks. Thus, we tabulate the fraction of simulation runs for which more multiplexing is associated with more diffusion.

In Figure 6 we plot the fraction of simulation runs where more multiplexing leads to more diffusion against the extent of diffusion in the network $p$. In panel A, we plot the results for simple diffusion. We find that higher multiplexing is consistently associated with lower diffusion levels, as in our theoretical results.

In panel B we see the nonmonotonicity from the countervailing forces in complex diffusion that we mentioned in Section 3.3. We also see a confirmation of the theoretical results. At low levels of diffusion, the steady state diffusion is increasing in multiplexing, and for high diffusion levels, the steady state diffusion is decreasing in multiplexing.