Download Diffusion and Contagion in Networks with Heterogeneous Agents

Diffusion and Contagion in Networks with
Heterogeneous Agents and Homophily∗
Matthew O. Jackson† and Dunia López-Pintado‡
Draft: September, 2012
Abstract
We study the diffusion of an idea, a product, a disease, a cultural fad, or a technology among agents in a social network that exhibits segregation or homophily (the
tendency of agents to associate with others similar to themselves). Individuals are distinguished by their types - e.g., race, gender, age, wealth, religion, profession - which,
together with biased interaction patterns, induce heterogeneous rates of adoption or
infection. We identify the conditions under which a behavior or disease diffuses and
becomes persistent in the population. These conditions relate to the level of homophily
in a society and the underlying proclivities of various types for adoption or infection.
In particular, we show that homophily can facilitate diffusion from a small initial seed
of adopters.
Keywords: Diffusion, Homophily, Segregation, Social Networks.
JEL Classification Numbers: D85, D83 C70, C73, L15, C45.
1
Introduction
Societies exhibit significant homophily and segregation patterns across various characteristics including age, ethnicity, geography, profession, education level, etc.1 These different
∗
Jackson gratefully acknowledges financial support from the NSF under grants SES-0961481 and SES1155302, from grant FA9550-12-01-GRAPHS-FP-028 of the AFOSR and DARPA, and from Microsoft Research New England. López-Pintado acknowledges support from the Spanish Ministry of Science and Innovation (ECO2008-03883, ECO2011-22919) as well as from the Andalusian Department of Economy, Innovation
and Science (SEJ-4154, SEJ-5980) via the “FEDER operational program for Andalusia, 2007-2013”. We
thank Cody Brown, Anne van den Nouweland, Myrna Wooders, and the anonymous referees for helpful
comments and suggestions on an earlier draft.
†
Stanford University, Santa Fe Institute, and CIFAR
‡
Universidad Pablo de Olavide and CORE, Université catholique de Louvain.
1
For background on homophily and some of its consequences, see McPherson et al. (2001) and Jackson
(2008), and for some background on how it can affect the empirics of diffusion processes see Aral, Muchnik,
and Sundararajan (2009).
1
groups can also have different susceptibilities towards adopting a new product or becoming
infected with a disease. For example, young children in day care may be highly susceptible
to certain infections and also have high rates of interaction with other young children, but
less interaction with older children. Adults of a given age range may be more likely to read a
particular book, and may have specific interaction patterns with other adults based on their
profession, religion, and education. How do such biases in interactions combined with different groups’ susceptibilities to contagion affect the adoption of products, spread of diseases,
spread of ideas, and other diffusion processes? For example, how does the diffusion of a new
product that is more attractive to one age group depend on the interaction patterns across
age groups? How does the answer depend on the differences in preferences of such groups,
their relative sociabilities, and biases in the interactions?
We answer these questions by analyzing a general model of diffusion that incorporates
a variety of previous models as special cases, including contagion processes studied in the
epidemiology literature such as the so-called SIS model (e.g., Bailey 1975, Pastor-Satorrás
and Vespignani, 2001), as well as interactions with strategic complementarities, such as in
the game theoretic literature and network games (e.g., Galeotti et al., 2010).2 Our model
incorporates types of individuals who have different preferences or proclivities for adoption
or infection, as well as biases in interactions across types.
In particular, we examine whether or not diffusion occurs from a very small introduction
of an activity in a heterogeneous and homophilous society. We first concentrate on the focal
situation with only two types of agents. Within this case, the most interesting scenario turns
out to be one where one type would foster diffusion and the other would not if the types
were completely segregated. In that scenario, we show that homophily actually facilitates
diffusion, so that having types biased in interactions towards their own types can enhance
diffusion to a significant fraction of both types. Having a higher rate of homophily, so that a
group is more introspective, allows the diffusion to get started within the group that would
foster diffusion on its own. This can then generate the critical mass necessary to diffuse the
behavior or infection to the wider society. In contrast, societies exhibiting less homophily
can fail to foster diffusion from small initial seeds.
We then move to the general case of many types. Our main characterization theorem
generalizes the features from the two-agent case, showing that diffusion relates to a condition
on the largest eigenvalue of an interaction matrix which tracks the initial adoption rates of
various types of individuals, that is, their adoption rates from small initial seeds. Again,
we show that homophily can facilitate diffusion, showing that a sufficient condition is that
some type (or group of types) that would adopt or become infected on its own is sufficiently
homophilous to give the diffusion a toehold. We discuss how this extends the intuitions from
the case of two types.
The most closely related work to ours is by Reluga (2009) and Galeotti and Rogers
2
For background on diffusion in networks see Newman (2002), Jackson and Yariv (2005, 2007, 2010),
López-Pintado (2006, 2008, 2012), Jackson and Rogers (2007), among others.
2
(2012). We take as given the susceptibility of each population to infection/adoption. In
some important applications this is not fixed, but can depend on things like vaccination
decisions. The populations can then have different incentives to immunize, which is the
main focus of Reluga (2009) and Galeotti and Rogers (2012). Their analyses apply to specific
settings (two equal-sized populations that are homogeneous within, and SIS dynamics) to
allow them to solve for endogenous immunization strategies; and thus there is little overlap
between our analysis and those papers except for that special case. An interesting direction
for further research would be to bridge the works to see how immunization decisions depend
on the diversity within populations, the differences in population sizes, and the number of
populations, and so forth.
Another related work is by Gómez-Gardeñes, Latora, Moreno, and Profumo (2008) who
study the diffusion of sexually transmitted disease in a heterosexual population. They examine a two-type SIS model, with a purely bipartite network so there is no interaction within
populations (i.e., between members of the same type). As we are interested in a general class
of diffusion processes, we consider a wider range of interaction patterns (allowing for any
mixing probabilities within and across populations), an arbitrary number of populations,
and more general contagion processes than the SIS case; and nest their setting as a special
case. Perhaps most importantly, we find that homophily is a critical determinant of diffusion,
something that is precluded in a bipartite network.
2
The Model
Consider a behavior (or disease) spreading in a heterogeneous population. Our aim is to
answer the question: If we start with a small seed of adopters (or infected agents), will the
behavior spread to a significant fraction of the population and become endemic?
In order to answer this question we approximate what is truly a random process on a finite
population by a deterministic process on a continuum population (essentially employing a
mean-field approximation), and all of our results come from this stylized model and so are
only suggestive of what might happen in a random and finite reality.
We begin with an overview of the model before providing additional details below.
2.1
Infection/Adoption States
Any individual can be in one of two states: 0 or 1 at any point in time - say, infected or not,
or having adopted a new technology or not, etc. Individuals may change states from time
to time or may stick to one state. An individual’s state is influenced by the individuals with
whom that individual has contact. The more individuals whom a given individual meets who
are in state 1, the more likely that individual is to end up in state 1. The more individuals
whom a given individual meets who are in state 0 the more likely that individual is to end
up in state 0.
3
2.2
Types and Degrees of Individuals
Individuals are distinguished by their characteristics, for instance: age, gender, profession,
wealth, and so forth; all of which can affect the individual’s likelihood to be in state 0 or 1
and to transition from one state to the other. For example, it may be that the elderly are
more susceptible to a particular disease, or it may be that teens of a certain ethnicity are
more likely to adopt a new product, or that adults with certain political leanings are more
likely to go to a particular documentary film. All of the relevant characteristics are captured
by a finite set of m ≥ 1 types: {1, 2, ...m}. So, type i might indicate males aged 30 to 50
living in a particular region, having a particular education level, religion, and so forth.
The continuum of agents N = [0, 1] is thus partitioned by the types, and we let n(i) ∈
[0, 1] denote the fraction of agents who are of type i.
In addition to types, individuals also are distinguished by the rate at which they have
interactions with others in the population. An individual’s degree d indicates how many
other individuals that person will have contact with and potentially be influenced by per
unit of time.
There can be heterogeneity in terms of degrees across types. For example some individuals
of type i might be more social than others and so it might be that some have d = 10
interactions while others have d = 5 interactions per unit of time. The distribution of
different degrees among all individuals of type i ∈ {1, . . . m} is described by the distribution
Pi , with Pi (d) indicating the fraction of individuals of type i who have d interactions per unit
of time. The distributions Pi and Pj may differ from each other, so for instance, teenagers
might have, say, a higher mean and variance in degrees compared to the elderly.
2.3
The Random Meeting Process
Per unit of time an individual of type i and degree d will meet d other individuals who might
influence that individual. The rate at which i meets agents of other types is described by
the matrix


π 11 . . . π 1m

..  .
Π =  ...
...
. 
π m1 . . . π mm
In particular, π ij is the probability that an agent of type i meets an agent of type j in
m
X
any given meeting. So,
π ij = 1.
j=1
The meeting process is allowed to be directional; i.e., one agent meeting (paying attention
to) another does not necessarily imply that the second pays attention to the first.3 For
instance, a child may emulate an adult, without the reverse occurring. So, although we use
the term “meeting,” the interaction need not be reciprocal.
3
Various media, including the internet, enable individuals to observe and follow others without reciprocation.
4
Of course, a special and relevant case is one where the interaction is mutual. If so, then
there are some constraints required on the meeting process as explained in Footnote 5.
We assume that Π is a primitive matrix (so that Πt > 0 for some t). This ensures that
there is at least some possibility for an infection that starts in one group to reach any other,
as otherwise there are some groups that are completely insulated from some others.
So, in a given period, an agent of type i with degree d expects to meet dπ ij agents of
type j. It can matter what the degrees of those agents of type j are. For example, meeting
a highly social agent of type j might lead to a higher chance that the type j individual is
“infected” than meeting a less social agent of type d. This can then affect whether i becomes
infected.
When meeting an agent of type j, the conditional probability that the agent met has
degree d is given by
Pj (d)wj (d),
where wj (d) is a weighting factor. For simplicity, we assume that wj (d) > 0 for all j and d
such that Pj (d) > 0.
If there is no weighting by degree, then an agent equally samples all agents of type j and
wj (d) = 1, and so the probability of meeting an agent of type j with degree d would simply
be Pj (d).4
Other values of the weighting function would represent a meeting process where there is
some correlation between the number of agents met by an agent (his/her degree) and the
number of agents who meet this individual. One relevant case is the one where meetings are
proportional to how social the agents are. That is, wj (d) = d/E[dj ]. This condition covers
cases in which meetings are reciprocal.5
2.4
The Infection or Adoption Process
As mentioned above, an agent is in one of two states {0, 1} at any point in time. We refer
to an agent in state 1 as being “infected” and an agent in state 2 as being “susceptible.”
4
This would usually require a directed meeting process, since the agents of type j would have equal
probabilities of being met, but would have different numbers of meetings, d, that they undertook themselves.
5
In such a context, certain constraints on the parameters of the model would be required. For example,
the number of interactions from type i to type j in a unit of time should coincide with the number of
interactions from type j to type i. Therefore, the following balanced equations should hold:
n(i)E[di ]π ij = n(j)E[dj ]π ji
for all i, j ∈ {1, ..., m}. If Π is symmetric then the conditions reduce to
n(i)E[di ] = n(j)E[dj ]
for all i, j ∈ {1, ..., m}. By varying the population size it is feasible to obtain the reciprocal meeting process
associated with any given profile of degree distributions and a symmetric interaction matrix Π (simply by
setting n(i) to be proportional to 1/E[di ] for each i). For additional details and references for random
meeting processes on a continuum, see the appendix of Currarini, Jackson and Pin (2009).
5
Agents’ types and interaction patterns determine their proclivities to become infected. A
susceptible agent of type i and degree d becomes infected at a rate described by a function
fi,d (a), where a is the number of agents whom she meets who are infected. The reverse
process, by which an infected agent of type i and degree d becomes susceptible happens at
a rate described by a function gi,d (a).
The functions fi,d (a) and gi,d (a) are the primitives of the diffusion process and are assumed
to satisfy some conditions. Essentially, the chance that a susceptible individual becomes infected should be nondecreasing in the number of infected individuals he or she meets, and
conversely the chance that an infected individual becomes susceptible should be nonincreasing in the number of infected individuals that he or she meets. Specifically, we assume the
following:
A1 There exists δ > 0 such that fi,d (0) = 0 and gi,d (0) = δ for each i and d: a susceptible
agent does not become infected unless she meets at least one infected agent, and all
types of agents become susceptible at some rate δ if they only meet susceptible agents.
A2 fi,d (a) is non-decreasing in a for each i and d: the rate at which susceptible agents
become infected is non-decreasing in the number of infected agents met.
A3 For each i fi,d (1) > 0 for some d such that Pi (d) > 0: for each type of agent there exists
some degree such that the rate at which susceptible agents with such a degree become
infected is positive when they meet at least one infected agent.
A4 gi,d (a) is non-increasing in a for each i and d: the transition rate from infected to
susceptible is non-increasing in the number of infected agents met.
Notice that A1 implies that the state where all agents are susceptible will be a stationary
state of the dynamics: if nobody is infected then nobody will become infected. Analyzing
the stability of the fully susceptible state is the main objective of the paper. A2 and A4
simply describe standard properties of a contagion (or coordination) process. Finally, A3
provides a technical requirement which will be necessary for the proof of Theorem 2 below.
This general model of diffusion embodies many different models, including those based on
best-response dynamics of various games (possibly with trembles) as well as epidemiological
models. Here are just few prominent examples of processes that are admitted:
• Susceptible-Infected-Susceptible Processes (an SIS diffusion process): The so-called
SIS model is a basic one used by the epidemiology literature to describe infectious
diseases spreading in a population (e.g., Bailey 1975, Pastór-Satorrás and Vespignani,
2000, 2001). In that model, each susceptible individual of type i becomes infected at
rate ν i > 0 times the number of infected individuals that he or she meets. Conversely,
at a rate δ > 0 an infected individual recovers and becomes susceptible again (where
for technical reasons, A3, we assume that this backward transition is independent of
6
type).6 The diffusion functions in this model are then fi,d (a) = ν i a and gi,d (a) = δ.
The crucial parameters of the SIS model are the relative spreading rate, λi = νδi , which
measure how infectious the disease is in terms of how easy it is to contract compared
to the rate at which one recovers.
• Best response dynamics by (myopic) agents who care about the relative play of neighbors (a relative threshold diffusion process): In such a model, agents choose
between technologies (e.g, new versus old, Windows versus Mac, etc.). The decision
depends on the behavior of the people that the individual meets. If more than a fraction qi are choosing 1, then the individual prefers to take action 1 and if less than a
fraction are choosing 1, then the individual prefers to take action 0. This adoption
process leads to the following diffusion functions: fi,d (a) = ν if ad ≥ qi and fi,d (a) = 0
otherwise, and gi,d (a) = δ if ad < qi and gi,d (a) = 0 otherwise, where qi ∈ [0, 1] is a
threshold characterizing the relative costs and benefits of action 1 versus 0. Here where
ν > 0 and δ > 0 are rates at which the agent pays attention to the neighbors per unit
time.7
• Imitation dynamics when a neighbor is chosen uniformly at random (an imitation
diffusion process): Again, as an example, consider the adoption of a new technology.
An agent of type i revises her strategy with a probability ν i > 0. She picks one of the
people whom she meets (uniformly at random) and imitates that person’s behavior.
Thus, fi,d (a) = ν i ad , and analogously, gi,d (a) = δ(1 − ad ) (where for technical reasons,
A3, we assume that this backward transition is independent of type).
6
The SIS model allows a recovered person to catch the disease again. This is the case for some sexually
transmitted diseases and is only an approximation for other infectious diseases like the standard flu where
new strains appear over time. Moreover, this is also true for diffusion of computer viruses or the diffusion of
many behaviors (e.g. choosing between Mac or Windows systems), where one may repeatedly change over
time.
7
One can model this by assuming that an agent of type i (player 1 in the table) obtains the following
payoffs when interacting with a neighbor (player 2 in the table)
2
1
0
ci
fi
ei
bi
1
1
0
(1)
where 1 stands for adoption of the new technology and 0 no adoption. Individuals have incentives to
coordinate their behavior and thus ci > fi and bi > ei . The payoff of playing 0 or 1 for a player of type i
in a given period is the sum of the payoffs obtained in each interaction. At a certain rate ν > 0 an agent
of type i currently choosing 0 revises her strategy. Then she selects the action that maximizes her benefits
given the actions of the people she meets (a myopic-best response). Therefore, if at time t the proportion of
i
met agents choosing 1 is higher than qi = ci −fbii−e
+bi −ei she switches to 1, otherwise she continues to choose 0.
Similarly, if she was choosing 1, she revises her strategy at a rate δ > 0 and switches to 0 if and only if the
i
proportion of agents choosing 1 is lower than qi = ci −fbii−e
+bi −ei . Again, the continuous time approximation is
for one where the agent would meet d other agents in discrete periods.
7
2.5
Individual Transitions from State to State
Let us now describe the rate at which an individual is expected to transition from being
susceptible to infected or vice versa, per unit of time.
Let ρi,d (t) denote the frequency of infected agents at time t among those of type i with
degree d. Thus,
X
ρi (t) =
Pi (d)ρi,d (t)
d
is the frequency of infected agents at time t among those of type i, and
X
ρ(t) =
n(i)ρi (t)
i
is the overall fraction of infected agents in the population at time t.
Let rate0→1
i,d (t) be the rate at which a susceptible agent of type i and with degree d
1→0
becomes infected at time t,8 whereas ratei,d
(t) stands for the reverse transition.
In order to compute these transition rates we must calculate first the probability that an
agent of type i has of meeting an infected agent, denoted e
ρi (t). It is straightforward to see
that
X
X
Pj (d)wj (d)ρj,d (t).
(2)
e
ρi (t) =
π ij
j
d
Given e
ρi (t) then
rate0→1
i,d (t)
=
d
X
fi,d (a)
d
a
gi,d (a)
d
a
e
ρi (t)a (1 − e
ρi (t))(d−a)
a=0
and
rate1→0
i,d (t)
=
d
X
e
ρi (t)a (1 − e
ρi (t))(d−a) .
(3)
a=0
These calculations are based on the expected number of meetings with infected agents given
d draws from a binomial distribution with probability e
ρi (t) of each draw being infected.9
2.6
Steady States and Dynamics
In order to keep track of how diffusion or infection occurs, we analyze a continuous time
dynamic, where at any given time t ≥ 0 the state of the system keeps track of the agents
who are infected and susceptible.
8
The t proxies for the state of the system at time t, so captures the various infection rates (ρj,d (t))j,d of
those who i will be expected to meet.
9
One could further approximate the system by evaluating f and g at the expected values rather than
taking their expectations: so using fi,d (de
ρi (t)) and gi,d (de
ρi (t)) in place of equations (3). These are the same
when f and g are linear in a (e.g., for the SIS model and imitation diffusion processes), but otherwise differ.
8
The diffusion dynamics are described as follows:
dρi,d (t)
0→1
1→0
= (1 − ρi,d (t))ratei,d
(t) − ρi,d (t)ratei,d
(t).
dt
(4)
The first expression on the right hand side tracks the increase in the level of infected agents
of type i and degree d due to the newly infected agents out of those who were susceptible,
and the second expression tracks the infected agents who have now changed to become
susceptible.
dρ (t)
A steady state is defined by i,d
= 0 for all i and d, which implies that a steadydt
10
state level ρi,d (t) is independent of time. Solving from equation (4) leads to the following
necessary condition for a steady state:
ρi,d
0→1
ratei,d
=
.
0→1
1→0
ratei,d
+ ratei,d
(5)
If we specify the rates e
ρi (t) for each type i, then this determines the rates of transition
under (3). This in turn, leads to a level of ρi,d for each i, d under (5) that would have to
hold in steady state, which in turn determines the rates at which infected agents would be
met under e
ρi (t). Thus, replacing equation (5) in equation (2) we find that a steady state
corresponds to a fixed-point as follows:
e
ρi = Hi (e
ρ1 . . . , e
ρn ),
where
Hi (e
ρ1 , . . . , e
ρn ) =
X
j
π ij
X
Pj (d)wj (d)
d
(6)
0→1
ratej,d
0→1
1→0
ratej,d
+ ratej,d
The previous system of equations implicitly characterizes the steady states, since by
solving for e
ρi we can easily find the fraction of infected agents of each type ρi and ultimately
the overall fraction of infected agents ρ.
2.7
Diffusion or Contagion from a Small Seed
We now consider the following question which is the central focus of our analysis: If we start
with a small fraction of infected agents, would a significant fraction of the population(s)
eventually become infected? In other words, we determine the conditions that lead to the
diffusion to a significant fraction of the population when there is a small initial perturbation of
an initial state in which everybody is susceptible; so starting from (ρ1 , . . . , ρn ) = (0, . . . , 0).11
10
In the case of best-response dynamics, steady states of the system correspond to Nash equilibria of the
game where players condition behavior on both their type and degree and those whom they meet from the
population in each period. In other cases, as for instance the SIS model, individuals display recurrent phases
of infection and susceptibility, but the system has a constant average infection rate.
11
The question of moving away from the population all being infected is analogous, swapping notation
between 0 and 1 throughout the model, and noting the reversed roles of f and g and given appropriate
variations of A1-A4.
9
e = 0 (and ρ = 0) is a steady-state, exploring the behavior of ρ
e near ρ
e=0
Given that ρ
e = 0 is a stable steady state.
under the dynamics described by (4) is a check of whether ρ
e = 0 is not stable, it is then easy to see that there is another (strictly
In cases where ρ
positive) steady-state given the monotonicity of the dynamic, given fixed-point arguments
applied to monotone correspondences on lattices (e.g., extensions of Tarski’s Theorem). 12,13
e = 0 is not stable, there will exist a steady state that has significantly positive
Thus, if ρ
rates of infection for all types, even if some of them would not have admitted any positive
steady states of infection in isolation.14
From (4) and (2) it follows that
X
de
ρi (t) X
0→1
1→0
=
π ij
Pj (d)wj (d) (1 − ρj,d (t))ratej,d
(t) − ρj,d (t)ratej,d
(t) .
dt
j
d
e=0
Note that by (3) and A1 it follows that near ρ
rate1→0
j,d (t) = δ
(7)
15
and (1 − ρj,d (t))rate0→1
ρj (t)d.
j,d (t) = fj (d, 1)e
(8)
Substituting from (8) into (7) we obtain
X
de
ρi (t) X
Pj (d)wj (d) fj (d, 1)e
ρj (t)d − ρj,d (t)δ .
=
π ij
dt
j
d
Then using (2), we rewrite this as
X
de
ρi (t) X
=
π ij
Pj (d)wj (d)dfj (d, 1)e
ρj (t) − e
ρi (t)δ.
dt
j
d
12
(9)
The monotonicity of the dynamics can be shown by checking that Hi (e
ρ1 , ..., e
ρ1 ) is nondecreasing in
e
ρj for each i and j. First note that fi,d (a) is nondecreasing in a (A2) and gi,d (a) is non-increasing in a
1→0
ρj , since these are
(A4). Then from (3), the rate0→1
j,d (t) is nondecreasing, and ratej,d (t) is nonincreasing, in e
expectations over the corresponding nondecreasing and nonincreasing fj and gj functions, respectively, taken
under binomial distributions with parameters e
ρj . The monotonicity of H then follows from the definition of
1→0
Hi , since the last fraction is then nondecreasing in e
ρj given that rate0→1
j,d (t) is nondecreasing and ratej,d (t)
is nonincreasing.
13
If H(ε, . . . , ε) > (ε, . . . , ε), then since H is nondecreasing and maps the compact space [ε, 1]m into itself,
there exists a (nonempty) complete lattice of fixed points in the set [ε, 1]m by Tarski’s Theorem. We establish
that the failure of stability of 0 implies that H(ε, . . . , ε) > (ε, . . . , ε) for some ε > 0 via Lemma 1.
14
One might also wonder whether there are positive steady states that are stable themselves. That can
depend on properties of H. For a more detailed discussion of that issue, see Jackson (2008, Chapter 9, Section
7). Even if there are no other stable steady states, the process would still be expected to move throughout
the positive region. Thus, if 0 is not a stable steady state, then regardless of the stability of positive steady
states, the process would live largely in the positive region, given any occasional perturbations.
15
e = 0. To
Also note that (1 − ρj,d (t)) and (1 − e
ρj (t))d−1 are both near 1 in small neighborhoods of ρ
be careful, we assume that there is an upper bound on degree which is not automatically implied given our
continuum of agents setting, but would be implied in any finite setting.
10
Rewriting (9), we have derived the following approximation of
e = 0:
near ρ
de
ρ(t)
e]δ
= [Ae
ρ−ρ
dt
where


π 11 x1 . . . π 1m xm


..
..
A=

.
...
.
de
ρ(t)
dt
in a neighborhood
π m1 x1 . . . π mm xm
and
xi =
X
Pi (d)wi (d)d
d
fi,d (1)
.
δ
The term xi is a nicely interpretable factor. It is the relative growth in activity due to
type i, but adjusted by the relative rates at which type i’s will be met by other agents (so
weighted by degrees according to wi (d)).
Note that if when we start with some vector of ρj ’s near 0 (so our approximation is
correct), but with positive entries, then we end up with a new vector that is larger than the
e]δ > 0.
starting vector, then it must be that 0 is an unstable solution, as then deρdt(t) = [Ae
ρ−ρ
Definition 1 There is diffusion from a small seed if and only if for any small ε > 0, there
exists some v such that 0 < vi < ε for all i and Av > v.
Thus, diffusion from a small seed requires that beginning from any small fraction of
initially infected agents the “dynamics” lead to a larger fraction of infected agents.
We remark that if 0 is unstable relative to some small initial seed v > 0, then it is
e > 0. That is, if Av > v, then for any v
e>0
unstable relative to any small initial seed v
t
e>v
e. Furthermore, if there is no diffusion with a particular
there is some t such that A v
small initial distribution, then there will be no diffusion with any other initial distribution.
The next Lemma formalizes this argument.16
Lemma 1 The condition for the diffusion from a small seed is independent of the distribution
e > 0 there
across types of the initial seed. That is, if Av > v for some v > 0, then for any v
t
e>v
e.
is some t such that A v
3
3.1
Analysis
Two Types
We begin with the analysis of two types.
16
This result is partly an artifact of the continuous model approximation. For an analysis of the importance
of the specifics of initially activated agents, see Banerjee, Chandrasekhar, Duflo and Jackson (2012).
11
For this two-type illustration, we stick with a setting where π 11 = π 22 = π, so that there
is a symmetry in how introspective groups are in terms of their meetings, and the more
general case is handled by Theorem 2.
1−x1 x2
Theorem 1 Let π 0 = x1 +x
. Diffusion occurs if and only if one of the following
2 −2x1 x2
conditions hold:
1) x1 x2 > 1 or
2) x1 x2 ≤ 1 and π > π 0 .17
The proof of Theorem 1 appears in the Appendix.
So, no diffusion occurs when x1 x2 ≤ 1 and π ≤ π 0 so that own-type interactions are
sufficiently low.18
The next corollary presents straightforward consequences of Theorem 1.
Corollary 1 In the two-type setting
1) If diffusion occurs within each type when isolated, then it would also occur when there
is interaction among the two.
2) If diffusion does not occur among either of the types when isolated, then it would not
occur when there is interaction among the two.
3) If diffusion would occur among only one of the types when isolated, then it would occur
among the entire population if homophily is high enough.
To see Corollary 1 first note that if there is only one type of agent in the population
then the condition for diffusion established by Theorem 1 reduces to the standard condition
of x > 1. Therefore, diffusion occurring within each type when isolated corresponds to
having x1 > 1 and x2 > 1. Those conditions in turn establish part 1) of the corollary as a
consequence of part 1) of Theorem 1. If, on the contrary, diffusion does not occur among
either of the types when isolated, then x1 < 1 and x2 < 1. Straightforward calculations show
that then the condition for diffusion stated in part 2) of Theorem 1 cannot be satisfied for
any value of π ∈ (0, 1). The last part of the corollary follows vacuously if x1 x2 > 1, and
otherwise diffusion occurs if π exceeds π 0 , establishing the claim.
If we consider, for example, the SIS model described earlier we have another corollary of
E[d2 ]
de1 λ1 de2 λ2
Theorem 1. Let π 0 = de λ +1−
, where dei = E[dii ] .
de λ −2de λ de λ
1 1
2 2
1 1 2 2
Corollary 2 Diffusion occurs from a small seed in the two-type SIS model if and only if
one of the following holds:
1) λ1 λ2 > de 1de or
1 2
2) λ1 λ2 ≤ de 1de and π > π 0 .
1 2
17
Note that the second condition implies that either x1 > 1 or x2 > 1.
This extends a finding obtained by Gómez-Gardeñes, Latora, Moreno, and Profumo (2008) for the
two-type SIS model.
18
12
The condition for diffusion in the standard (homogeneous) SIS model is λ > d1e (e.g.,
see Pastor-Satorrás and Vespignani, 2001). Thus, we see how this generalizes in the above
theorem. In particular, the absence of an epidemic threshold for scale-free networks, a wellknown result for the SIS model, extends to the heterogeneous case. In particular, if at
least one of the two degree distributions is scale-free (e.g., Pi (d) is proportional to d−γ , for
γ ∈ (2, 3)), then regardless of how small the infectious rates λ1 and λ2 are, as long as they
are positive the disease will spread and become endemic in the population.19 Notice that
the condition for diffusion depends not only on the product of the two infection rates and
the two degree distributions (in particular, on the ratio between the second and first order
moments of the distributions) but also on the level of homophily in the population.
3.2
The General Case with Many Types
We remark that since xi > 0 for all i (under A1-A4), and since Π is nonnegative and
primitive, it follows that A is primitive and thus At > 0 for some t.
We can now state the following result, which generalizes the two-type result to many
types.
Theorem 2 Diffusion occurs if and only if the largest eigenvalue of A (denoted by µ) is
larger than 1.
The proof of Theorem 2 appears in the appendix.
Corollary 3 extends Corollary 1 to the m-type case.
Corollary 3
1) If diffusion from a small seed occurs within each type when isolated, then it would also
occur when there is interaction among types.
2) If diffusion from a small seed does not occur for any of the types when isolated, then
it would not occur when there is interaction among them.
3) If there is some type for which π ii xi > 1, then there is diffusion from a small seed.
P
4) If there is a subset of types S ⊂ T such that j∈S π ij xj > 1 for each i ∈ S, then there
is diffusion from a small seed.
We first explain why 1) holds, as 2) is a simple variation. If diffusion occurs within
each type when isolated then xi > 1 for all i and therefore A > Π (where this indicates a
higher entry for non-zero entries, and ultimately a higher value for all entries when raised
to a sufficient power given that both matrices are primitive). It follows that the largest
eigenvalue of A is larger than 1 (since the right-hand side matrix is a stochastic matrix and
thus has a largest eigenvalue of 1), and the result then follows from Theorem 2.
This is due to the fact that e1 = 0 in a scale-free network. That presumes an unbounded distribution.
di
If the distribution is truncated at some maximal degree, then this expression will be positive, but can be
arbitrarily close to 0 depending on the maximal degree.
19
13
Next let us explain why 3) and 4) are true, and then discuss the intuition. 3) is clearly
P
a special case of 4), so let us discuss why 4) is true. Given that j∈S π ij xj > 1 for each
i ∈ S, it follows that for any positive vector u: [Au]i is greater than minj∈S uj for each
i ∈ S. Therefore, minj∈S [Au]j > minj∈S uj , and so it must be that if u is the eigenvector
corresponding to the maximum eigenvalue,20 then Au > u and so the eigenvalue is larger
than 1.
1) and 2) of the corollary are fairly intuitive results. Note that in case of just one
population, then x > 1 is the condition that characterizes instability of (diffusion from) no
activity. Thus, if all populations are such that they would experience diffusion from a small
seed if isolated, then regardless of the interaction pattern there will be diffusion; and similarly
if none of them would experience diffusion in isolation, then there cannot be diffusion when
they interact.
The less obvious cases are 3) and 4), which show that if some type or group of types
has enough interaction with itself to get diffusion going, then diffusion among the entire
population will occur. Again, these emphasize the role of homophily in enabling diffusion
(infection) from a small seed: if there is some group of types that interacts within itself in
a manner sufficient to enable diffusion among that group, then a toehold can be established
and diffusion will occur from a small seed.
Another corollary is that if populations are similar so that they have the same infection
properties near 0 (i.e., xi = xj = x for all i and j), then diffusion properties are determined
by whether this growth rate is bigger or smaller than 1.
Corollary 4 If xi = xj = x for all i and j, then there is diffusion from a small seed if
and only if x > 1.
This corollary then emphasizes that in order for the homophily and particular patterns
of interaction to matter, it must be that types are not just heterogeneous in their interaction
(the Π matrix), but also in their infection proclivities. If they all have similar infection
proclivities, then the particular details of who interacts with whom do not affect diffusion
from a small seed.
The proof of this corollary is straightforward. Note that in this case A = xΠ It follows
that the largest eigenvalue of A is larger than 1 if and only if x > 1 since Π is a stochastic
matrix and has a largest eigenvalue of 1.
The less obvious cases are such that there are some types who would experience diffusion
on their own, while others would not. Then as already illustrated for the two-type case,
some subtle conditions ensue. Beyond the sufficient condition mentioned above that there
be sufficient homophily for an infection to take hold within some type, more complicated
patterns among a number of groups can also possibly lead to diffusion from a small seed.
20
Again, recall that A is primitive and thus has a strictly positive eigenvector corresponding to its largest
eigenvalue.
14
Perhaps most importantly, we have seen that contrary to a simple intuition, homophily
can enhance diffusion, and so the welfare implications of homophily can depend on whether
diffusion is desired (e.g., in the case of an efficient new technology) or not (e.g., in the case
of a disease) and whether the populations would support diffusion in isolation.
4
Concluding Remarks
Much of the study of how social network structure impacts diffusion has focused on the
distribution of degrees.21 Here we have addressed the effect of homophily on whether or not
diffusion takes place, something which despite its importance has received little attention
in the diffusion literature.22 As a first step to understanding the effect of homophily on
diffusion, in this paper we focused on a specific question; namely the spreading of a new
behavior when starting with a small initial seed.
The main insight from our analysis is that homophily can facilitate infection or contagion,
by allowing an infection to get a toehold in one population and subsequently spread. If
infection would already spread in each population in isolation, or in none of the populations
in isolation, then there can be no effect. The interesting case is then when some populations
would harbor infection in isolation and others would not. In this situation, homophily plays
an important role. If populations interact with each other sufficiently, then the population
that might harbor infection on its own might never reach an endemic state since they meet
each other too rarely for infection to get started. However, when there is sufficient homophily,
but still some interaction across groups, then infection can get started within one group and
then transfer to others. Just as an example, schools can be strong incubators for a variety
of diseases which then can spread beyond. Our results thus provide a variety of testable
hypotheses relating whether diffusion occurs to proclivities of various groups for infection
and relative interaction rates across groups.
There are other issues which are left for further work. For example, one could evaluate
the extent of diffusion as a function of the homophily level. Homophily can have conflicting
effects: although it can facilitate an initial diffusion, the ultimate effect on the diffusion can
go either way.
Examples of the different extents of the ultimate diffusion and how it can depend nonmonotonically on homophily are easy to construct as follows. Consider a situation with two
21
E.g., Pastór-Satorrás and Vespignani, 2000, 2001, Jackson and Rogers, 2007, López-Pintado, 2008,
Galeotti and Goyal, 2009, Galeotti et al., 2010.
22
An exception are the studies by Golub and Jackson (2012abc) which also examine the impact of homophily on some (very different) learning and diffusion processes. There are important differences between
our approach and theirs. We focus on generalizations of the SIS infection model and network games, whereas
Golub and Jackson (2012a) analyze models of diffusion based either on shortest paths communication, random walks, or linear updating processes. Second, the paper by Golub and Jackson (2012a) studies the
convergence time to the steady state, whereas we analyze whether there is or not convergence to a state with
a positive fraction of infected agents.
15
populations, one which would foster diffusion on its own, and another that would not, but
once infected would spread to a high level of infection. With no homophily, and appropriate parameters there is no diffusion, then as homophily increases to a middle range one
gets diffusion starting through via the more susceptible population and then spreading very
extensively, but then as homophily increase further, the diffusion is mostly confined to the
more susceptible population.
References
[1] Aral, S., Muchnik, L., and Sundararajan, A. (2009) “Distinguishing Influence-Based
Contagion from Homophily-Driven Diffusion in Dynamic Networks,” Proceedings of the
National Academy of Sciences, 106(51), 21544-21549.
[2] Bailey, N.T.J. (1975) The Mathematical Theory of Infectious Diseases and Its Applications, London Griffin.
[3] Banerjee, A., Chandrasekhar, A. , Duflo, E., and Jackson, M.O. (2012) “The Diffusion
of Microfinance,” Preprint: http://www.stanford.edu/∼jacksonm/diffusionofmf.pdf
[4] Currarini, S., Jackson, M. O., and Pin, P. (2009) “An Economic Model of Friendship:
Homophily, Minorities and Segregation,” Econometrica, 77(4), 1003 - 1045.
[5] Galeotti, A., Goyal, S., Jackson, M. O., Vega-Redondo, F., and Yariv, L. (2010) “Network Games,” The Review of Economic Studies, 77(1), 218-244.
[6] Galeotti, A., and Rogers, B. (2012) “Strategic Immunization and Group Structure,”
forthcoming in the American Economic Journal: Microeconomics.
[7] Golub, B. and Jackson, M.O. (2012a) “How homophily Affects the Speed of Learning
and Best Response Dynamics,” in the Quarterly Journal of Economics, Vol. 127, Iss. 3,
pp 1287 - 1338.
[8] Golub, B. and Jackson, M.O. (2012b) “Does Homophily Predict Consensus Times?
Testing a Model of Network Structure via a Dynamic Process,” in the Review of Network
Economics, Manuscript 1367, pp 1 - 28.
[9] Golub, B. and Jackson, M.O. (2012c) “Network Structure and the Speed of Learning:
Measuring Homophily Based on its Consequences,” in the Annals of Economics and
Statistics, 107/108 July-Dec.
[10] Gómez-Gardeñes, J., Latora, V., Moreno, Y., Profumo, E. (2008) “Spreading of Sexually transmitted Diseases in Heterosexual Populations,” Proceedings of the National
Academy of Sciences, 105(5), 1399-1404.
16
[11] Jackson, M. O. (2008) Social and Economic Networks, Princeton University Press: NJ.
[12] Jackson, M. O. and Rogers, B. W. (2007) “Relating Network Structure to Diffusion
Properties through Stochastic Dominance,” The B.E. Press Journal of Theoretical Economics, 7(1) (Advances), 1-13.
[13] Jackson, M. O. and Yariv, L. (2005) “Diffusion on Social Networks,” Economie Publique,
16(1), 69-82.
[14] Jackson, M. O. and Yariv, L. (2007) “Diffusion of Behavior and Equilibrium Properties
in Network Games,” The American Economic Review (Papers and Proceedings), 97(2),
92-98.
[15] Jackson, M. O. and Yariv, L. (2011) “Diffusion, Strategic Interaction, and Social Structure,” in the Handbook of Social Economics, edited by Benhabib, Bisin and Jackson,
North Holland.
[16] López-Pintado, D. (2006) ”Contagion and coordination in random networks,” International Journal of Game Theory, 34, 371-381.
[17] López-Pintado, D. (2008) ”Diffusion in complex social networks,” Games and Economic
Behavior, 62, 573-90.
[18] López-Pintado, D. (2012) ”Influence Networks,” Games and Economic Behavior, 75,
776 - 787.
[19] McPherson, M., Smith-Lovin, L., and Cook, J. M. (2001) “Birds of a Feather: Homophily in Social Networks,” Annual Review of Sociology, 27, 415-444.
[20] Newman, M.E.J. (2002) “The spread of epidemic disease on networks,” Physical Review
E, 66, 016128.
[21] Pastor-Satorras, R. and Vespignani, A. (2000) “Epidemic Spreading in Scale-Free Networks,” Physical Review Letters, 86(14), 3200-3203.
[22] Pastor-Satorras, R. and Vespignani, A. (2001) “Epidemic dynamics and endemic states
in complex networks,” Physical Review E, 63, 066117.
[23] Reluga, T.C., (2009) “An SIS Epidemiology Game with Two Subpopulations,” Journal
of Biological Dynamics, 3(5), 515-531.
17
Appendix
We present the proofs in an order different from the results in the paper, as some of the
proofs use another.
Proof of Theorem 2: As mentioned in the text, A is primitive since Π is primitive and
since A3 is satisfied implying that xi > 0.23 Thus, by the Perron–Frobenius Theorem (which
applies to primitive matrices) the maximum eigenvalue, denoted µ hereafter, is positive and
its corresponding eigenvector, denoted by u hereafter, is also positive.
We show next that the condition for diffusion from a small seed, or the instability of
e = 0, corresponds with the condition that the largest eigenvalue of A is larger than 1.
ρ
e = 0 is unstable. Note that if µ > 1 then
Let us first show that if µ > 1 then ρ
Aδu = µδu > δu.
Thus, picking small enough δ so that δui < ε for each i, satisfies the definition of diffusion
from a small seed with δu (or instability of 0).
To see the converse, first consider the case such that µ < 1. Given ε > 0 consider any v
such that 0 < vi < ε for all i. Suppose that Av > v. It then follows A(Av) > Av > v as
A is nonnegative and has at least one positive entry in each row. Iterating, it follows that
that At v > v for any t. However, choose δ such that δu > v. Given that A is is nonnegative
and has at least one positive entry in each row, and both vectors are positive, it follows that
Aδu > Av, and similarly that
At δu > At v.
Given our previous claim, this then implies that
At δu > v
for all t. However,
At δu = δµt u → 0
given that µ < 1, which is a contradiction.
To complete this part of the proof consider the case such that µ = 1. Consider ε > 0.
Consider any vector v such that vi < ε. Note that for any small enough δ > 0 the largest
eigenvalue of A − δI is less than 1. Thus, by the argument above, (A − δI)v is not greater
than v. Therefore, Av is not greater than v.
e = 0 is unstable if and only if the
Proof of Theorem 1: We have already shown that ρ
largest eigenvalue of matrix A is above 1. Let us now complete the proof by examining the
eigenvalue in the two-type case. The eigenvalues of a 2 × 2 matrix are easily computed.
Writing
a11 a12
A=
,
a21 a22
23
In fact, with two types A is a positive matrix since 0 < π < 1.
18
the largest eigenvalue of A is24
µ=
p
(a11 + a22 ) +
(a11 + a22 )2 − 4(a11 a22 − a12 a21 )
2
or equivalently
µ=
a11 + a22 +
p
(2 − a11 − a22 )2 − 4 + 4a11 + 4a22 − 4a11 a22 + 4a12 a21
.
2
Thus, µ is larger than 1 if and only if
a11 + a22
>1
2
(10)
− 1 + a11 + a22 − a11 a22 + a12 a21 > 0.
(11)
or
Given that a11 = πx1 , a22 = πx2 , a12 = (1 − π)x2 and a21 = (1 − π)x1 then conditions (10)
e = 0) occurs if and only if
and (11) imply that diffusion (i.e., instability of ρ
π>
2
x1 + x2
(12)
or
π(x1 + x2 − 2x1 x2 ) + x1 x2 − 1 > 0.
Case 1:
x1 +x2
2x1 x2
(13)
> 1. In this case, condition (13) is equivalent to
π>
1 − x1 x2
x1 + x2 − 2x1 x2
and therefore diffusion occurs in this case if and only if
π > min{
Case 2:
x1 +x2
2x1 x2
1 − x1 x2
2
,
}.
x1 + x2 − 2x1 x2 x1 + x2
(14)
< 1. In this case, condition (13) is equivalent to
π<
x1 x2 − 1
2x1 x2 − x1 − x2
and therefore diffusion occurs in this case if and only if
2
x1 x2 − 1
< π or π <
.
x1 + x2
2x1 x2 − x1 − x2
(15)
2
Case 3: x2x1 +x
= 1. In this case, condition (13) simplifies to x1 x2 > 1, and and therefore
1 x2
diffusion occurs in this case if and only if
2
< π or x1 x2 > 1
x1 + x2
24
Note that since A is primitive, its largest eigenvalue is real and positive.
19
(16)
Let us now show part (1) of Theorem 1.
2
Suppose that x1 x2 > 1 holds. Then x2x1 +x
can fall into any of the cases above. If it were
1 x2
1−x1 x2
greater than 1, then x1 +x2 −2x1 x2 < 0 which in particular by Case 1 and (14) implies that
there is diffusion for any π ∈ (0, 1). If it were equal to 1, then by Case 3, the result holds.
2
< 1 then Case 2 applies. In that case, referring to Figure 1, (x1 , x2 ) lies
If instead x2x1 +x
1 x2
above the upper-most curve,25 and it is clear that there would exist another profile (b
x1 , x
b2 )
such that x
b1 ≤ x1 and x
b2 ≤ x2 and which lies in the regions considered previously (that is,
1−x1 x2
where x1 +x2 −2x1 x2 ≤ 0). Therefore diffusion for (b
x1 , x
b2 ) occurs for all π ∈ (0, 1), which in
particular implies that for the larger case (x1 , x2 ) diffusion would also occur for all π ∈ (0, 1)
as the largest eigenvalue of a larger matrix is necessarily larger than the largest eigenvalue
of a smaller matrix.
x2
-
1
+
(x1+x2)/2x1x2=1
1/2
1
+
-
x1+x2=2
+
x1x2=1
x1
Figure 1: The relationship between the key expressions in the proof of Theorem 1.
2
Next, we show part (2) of Theorem 1. Suppose that x1 x2 ≤ 1. This implies that x2x1 +x
>1
1 x2
(see Figure 1) or else that x1 = x2 = 1 in which Case 3 applies and there cannot be diffusion.
2
Thus, let us analyze the situation where x2x1 +x
> 1 and Case 1 applies. Diffusion occurs
1 x2
1−x1 x2
2
2
if and only if π > min{ x1 +x2 −2x1 x2 , x1 +x2 }. Note that if x1 + x2 < 2 then x1 +x
> 1 and
2
1−x1 x2
therefore diffusion occurs if and only if π > x1 +x2 −2x1 x2 . If, on the contrary, x1 + x2 ≥ 2
1−x1 x2
2
then, it is straightforward to show that x1 +x
> x1 +x
which also implies that diffusion
2
2 −2x1 x2
1−x1 x2
in such a case occurs if and only if π > x1 +x
.
2 −2x1 x2
25
The relative positions of the curves are easily checked, and note the plus and minus signs that indicate
whether one is above or below 1 for the corresponding colored expression.
20
Proof of Lemma 1: Given the proof of Theorem 2, it follows that if Av > v for some
b. It follows that Aδu < Ab
v > 0 then µ > 1. Then, choose δ such that δu < v
v (since A is
nonnegative and has at least one positive entry in each row), and similarly that
b,
µt δu = At δu < At v
and the first expression is growing with µt .
21