The Network Structure of Unequal Diusion Eaman Jahani Department of Statistics University of California Berkeley

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The Network Structure of Unequal Diﬀusion

Eaman Jahani ∗

Department of Statistics, University of California, Berkeley

Dean Eckles

MIT Sloan School of Management

Alex “Sandy” Pentland

MIT Institute for Data, Systems and Society

September 15, 2022

Abstract

Social networks aﬀect the diﬀusion of information, and thus have the potential

to reduce or amplify inequality in access to opportunity. We show empirically that

social networks often exhibit a much larger potential for unequal diﬀusion across

groups along paths of length 2 and 3 than expected by our random graph models. We

argue that homophily alone cannot not fully explain the extent of unequal diﬀusion

and attribute this mismatch to unequal distribution of cross-group links among the

nodes. Based on this insight, we develop a variant of the stochastic block model

that incorporates the heterogeneity in cross-group linking. The model provides an

unbiased and consistent estimate of assortativity or homophily on paths of length 2

and provide a more accurate estimate along paths of length 3 than existing models.

We characterize the null distribution of its log-likelihood ratio test and argue that the

goodness of ﬁt test is valid only when the network is dense. Based on our empirical

observations and modeling results, we conclude that the impact of any departure

from equal distribution of links to source nodes in the diﬀusion process is not limited

to its ﬁrst order eﬀects as some nodes will have fewer direct links to the sources.

More importantly, this unequal distribution will also lead to second order eﬀects as

the whole group will have fewer diﬀusion paths to the sources.

Keywords: Stochastic Block Model, Assortativity, Diﬀusion Paths, Brokerage, Heteroge-

neous Edge Propensities

∗Authors would like to thank Matthew Jackson for his valuable comments and feedback and Tiago

Peixoto for his helpful insights and references.

arXiv:2210.11053v1 [stat.AP] 20 Oct 2022

1 Introduction

Diﬀusion of information in social networks determines who gets access to a valueable piece

of information, such as a new investment opportunity. The structure of the network plays

an important role in which individuals or groups receive the valuable information. Certain

network structures are more likely to keep a piece of information exclusive to one group, thus

leading to unequal diﬀusion. For example, if there are very few social links between people

of diﬀerent races, the information about a new employment opportunity that is generated

among one race might never reach individuals of the other race (Calv´o-Armengol and

Jackson, 2004). Many existing network models aim to explain the absence of diﬀusion from

one group to another through assortative mixing (Newman, 2003b). Assortative mixing,

or simply assortativity, captures the bias in forming edges with similar characteristics. It

is also referred to as homophily which simply means that attributes of nodes are correlated

across the edges. For example, in social networks individuals have a strong tendency to form

links with other people who are similar to them in terms of age, language, socioeconomic

status or race.

The stochastic block model (SBM) — along with its variants such as degree-correction

(Karrer and Newman, 2011) — deﬁnes an important class of these models that explicitly

account for assortative mixing in networks. SBM is a generative random network model

for modeling blocks or groups in networks. It has been widely used in computer science

and social sciences to model community structure in networks (Rohe et al., 2011; Holland

et al., 1983a; Anderson et al., 1992; Faust and Wasserman, 1992; Wasserman and Faust,

1989; Wang and Wong, 1987). In its original form, vertices in a network exclusively belong

to one of the Kgroups (or blocks) in the network. Each pair of vertices form an edge

independently of other edges or vertices. Edge formations between any pairs of two groups

are independent, identical and solely determined by the group membership of the pair of

vertices. If gi∈ {1,2, ..., K}corresponds to the group of vertex i, then a K×Kmatrix, P,

(a) (b)

Figure 1: Comparison of (a) a network with brokerage in which a disproportionate fraction

of cross-type edges are held by a small number of nodes versus (b) a similar network in

which cross-type edges are distributed more equally. Red and blue nodes correspond to two

diﬀerent groups or blocks. Corresponding nodes have the same degree and the number of

cross-type edges are the same in both networks, but there are 10 cross-type paths of length

2 of the form red-blue-blue in network (b) while there are only 8 such paths in network (a).

determines the edge formation probabilities between any pair of vertices. The probability

of an edge between any pairs iand jis the (gi, gj) element in the matrix, Pgi,gj.

This simple model can produce a variety of interesting network structures. For example,

an edge probability matrix in which diagonal entries are much larger than oﬀ-diagonal

entries produces networks with densely connected groups and sparse connections across

groups. The ability to model such community structure is the main reason SBM can

capture assortative mixing in a network. This has led to the popularity of SBM as one of

the main methods for community detection. SBM does so by generating random networks

that match the observed network in terms of the frequency of within-group and cross-group

edges. The ﬁtted model matches the observed assortativity or homophily in expectation.

SBM or its degree corrected version assume that within-group and cross-group edges

are distributed “uniformly” across all pairs: the existence of an edge between any two

pairs is identical to other similar pairs. In the case of degree-corrected SBM (DCSBM),

after conditioning on degree two nodes are similar in terms of their cross-group edge for-

mation. In reality, many real networks have heterogeneous propensities in edge formation

to various groups. In most cases, social networks exhibit a pattern of brokerage which

means cross-group edges are not distributed uniformly, instead a small subgroup of nodes

hold a disproportionate level of cross-group edges. Simmel (1950) was the ﬁrst to intro-

duce the concept of network brokerage in triadic relations. Burt (2009) later advanced our

understanding of brokerage by introducing the concept of “structural holes” between two

unconnected communities, across which brokers act as intermediary. These broker nodes

play an important role in connecting otherwise disconnected communities, moving infor-

mation between them, and acting as an intermediary for resource exchanges. Due to their

unique position in the network, brokers beneﬁt from various types of advantages, for ex-

ample access to diverse information or opportunities for arbitrage in exchanges. However,

these advantages to brokers might lead to some costs to other actors in the network or the

network as a whole.

Figure 1a provides a visual example of a network with brokerage in which a small number

of broker nodes have a higher propensity to form links with brokers of the out-group, hence

maintaining majority of cross-group edges. Figure 1b shows a similar network with less

brokerage which has more frequent cross-type paths of length 2 even though it has the

same degree distribution as the brokerage network 1a. While brokers play an integral role

in connecting otherwise disconnected communities, they can nevertheless act a bottleneck

by reducing the number of possible paths between any two groups when compared to a

similar network with cross-group ties uniformly distributed across the network. Because

brokers hold a disproportionate number of cross-group ties, they can constrain diﬀusion

of information from one group to another. In this paper, we argue that one needs to not

only look at homophily or assortativity on paths of length 1, but also on the extent of

assortativity of all possible diﬀusion paths of varying lengths to completely account for

unequal diﬀusion in networks. We then attempt to incorporate the heterogeneity in edge

propensities and in particular brokerage into class of Stochastic Block Models and show

that by doing so the model better explains unequal diﬀusion of information.

We show that while directly ﬁtting for assortativity on paths of length 1, SBM fails to

accurately capture assortativity on longer paths in real world networks. In the context of

random graphs, network brokerage occurs when a few nodes in the network have higher

probability to connect with an out-group than other in-group nodes. By incorporating

this heterogeneity into our models of random network and in particular SBM or degree-

corrected SBM, we show that the generative model can better match the assortativity along

longer paths and more generally cross-type diﬀusion in the observed network. In section 2,

we discuss SBM and some variant models and show that they consistently under-estimate

the observed assortativity on paths of length 2 in 56 school networks, even though these

models explicitly accounts for assortativity on paths of length 1. In section 3, we discuss

a general framework for Stochastic Block Models and develop variants which account for

node heterogeneity in brokerage and by doing so match assortativity on paths of length 1

and 2 in expectation. In section 4, we provide the results from ﬁtting the school networks

to our model and show that even though not explicitly modeled for, it closely matches

assortativity on paths of length 3. In section 5, we address the goodness of ﬁt for this new

model versus one that does not account for brokerage. We characterize the distribution

of the log likelihood ratio statistic and argue that the test is valid only if the network is

dense, which is often not the case for social networks.

In the remainder of this paper, we mostly focus on assortativity of path length 2 and 3

as opposed to longer paths. While diﬀusion as a general process can occur across paths of

any length, nevertheless in many scenarios, especially those that involve access a valuable

resource, diﬀusion mostly occurs along short paths. Therefore, while assortativities along

paths of length 2 and 3 do not provide an exact representation of diﬀusion assortativity,

we believe they nevertheless provide a simple and interpretable model that is applicable in

most social contexts.

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TheNetworkStructureofUnequalDiusionEamanJahani*DepartmentofStatistics,UniversityofCalifornia,BerkeleyDeanEcklesMITSloanSchoolofManagementAlex\Sandy"PentlandMITInstituteforData,SystemsandSocietySeptember15,2022AbstractSocialnetworksaectthediusionofinformation,andthushavethepotentialtoreduceorampli...

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