SMSM References

# Influencing dynamics on social networks without knowledge of network
microstructure

Matthew Garrod  Department of Mathematics, Imperial College London, London SW7
2AZ, UK  Marks and Spencer Group plc. Waterside House 35 North Wharf Road
London W2 1NW  Nick S. Jones  Department of Mathematics, Imperial College
London, London SW7 2AZ, UK  EPSRC Centre for the Mathematics of Precision
Healthcare, Department of Mathematics, Imperial College London, London SW7
2AZ, United Kingdom.

###### Abstract

Social network based information campaigns can be used for promoting
beneficial health behaviours and mitigating polarisation (e.g. regarding
climate change or vaccines). Network-based intervention strategies typically
rely on full knowledge of network structure. It is largely not possible or
desirable to obtain population-level social network data due to availability
and privacy issues. It is easier to obtain information about individuals’
attributes (e.g. age, income), which are jointly informative of an
individual’s opinions and their social network position. Using a statistical
mechanics based model of opinion formation, we show that it is possible to
effectively influence dynamics on networks without full knowledge of network
structure. In particular, influence strategies which use coarse grained
network data, external covariates and point estimates of the state can all
outperform a baseline model. Our work provides a scalable methodology for
influencing Ising systems on large graphs and the first exploration of the
Ising influence problem in the presence of ambient (social) fields. By
exploiting the observation that strong ambient fields can simplify control of
networked dynamics, our findings open the possibility of running public
information campaigns using insights from social network theory without costly
or invasive levels of data collection.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Background

Social network information is widely used for carrying out public information
campaigns. In a public health context, social network interventions have been
applied for mitigating HIV transmission heckathorn1999aids ;
rice2012mobilizing , diabetes management vissenberg2016impact ;
vissenberg2017impact ; vissenberg2017development  and reducing smoking
pechmann2017randomised ; tsoh2015social . Social network structure is also
relevant for understanding and mitigating polarisation in public sentiment
concerning topics such as climate change williams2015network ;
kaiser2017alliance ; kahan2012polarizing  and vaccine sentiment
schmidt2018polarization . This observation has motivated the development of
strategies for optimally influencing the spread of a contagion
kempe2003maximizing ; chen2010scalable ; wang2012scalable  and equilibrium
models for opinion dynamics lynn2016maximizing ; lynn2017statistical ;
hindes2017large ; lynn2018maximizing . However, sourcing population-level
social network information has substantial privacy, reproducibility and
generalizability concerns. Given this, we might ask – what level of
information is required to effectively influence opinion dynamics on networks?

There is a long tradition in the social sciences of using surveys to collect
data about people and their relationships wasserman1994social . These surveys
typically collect demographic coordinates such as age, gender, ethnicity,
income and education level. The multidimensional space spanned by peoples’
demographic attributes is commonly referred to as a _Blau space_
mcpherson2004blau ; genkin2018blaunet  after sociologist Peter Blau
blau1977macrosociological . The principle of homophily states that people are
more likely to share connections with others who share similar traits
mcpherson2001birds . Consequently, at the macroscale, the likelihood of two
individuals sharing a social tie will be a decreasing function of their
‘social distance’ within Blau space. We specify this probability using a
_connectivity kernel_ or connection probability function. Data from social
surveys can be used to infer these connectivity kernels hoffmann2018partially-
observed ; hoffmann2020inference  or estimate global properties of underlying
networks hoffmann2020inference ; godoy-lorite2020inference . This might be
achieved via ego network surveys smith2019continued ; smith2015global ,
collecting aggregated relational data mccormick2015latent ; breza2017using ;
breza2019consistently  or inference from behavioural data godoy-
lorite2020inference .

The Blau coordinates obtained from surveys will often be discrete or
discretized (e.g. age and income data might be divided into bins). In this
case, an appropriate modelling framework is one in which the network has
modular structure, so that nodes are more likely to share connections with
others in the same module. We will use the stochastic block model (SBM)
karrer2011stochastic ; holland1983stochastic ; faust1992blockmodels  in order
to model networks with modular structure.

In this paper, we explore whether it is feasible to influence binary state
dynamics on a network given knowledge of external covariates and a
connectivity kernel. We model opinion dynamics using the Ising model
castellano2009statistical ; acemoglu2011opinion . The Ising model provides a
parsimonious framework which includes network effects, noise and external
explanatory variables. It exhibits the metastability characteristic of many
real world systems hindes2017large , links to threshold models used in the
social sciences granovetter1978threshold  and fits within the economic
framework of discrete choice theory mcfadden2001economic ; sornette2014physics
. In the context of influencing large scale social systems the impact of
demographics can be encoded in ‘background’ or ‘ambient’ fields godoy-
lorite2020inference ; gallo2009parameter ; opoku2019parameter .

Ising systems on networks with modular structures have been studied widely in
the context of the _multi-species Curie-Weiss model_ fedele2013inverse ;
fedele2017inverse ; opoku2018multipopulation  (MSCW). This model is equivalent
to the Ising model on a weighted complete graph. MSCW models have been fit to
data concerning social outcomes including health screening campaigns
burioni2015enhancing , civil vs religious marriages, divorces and suicidal
tendencies gallo2009parameter ; gallo2009equilibrium  and educational
attainment opoku2019parameter . In these models, individuals are partitioned
into groups based on demographic attributes. In the MSCW model the graph is
assumed to be fully connected. Ising systems on non-fully connected systems
have been considered in the context of the Kernel Blau Ising model godoy-
lorite2020inference , coupled Barabási-Albert networks suchecki2006ising ;
suchecki2009bistable-monostable  and SBMs lowe2019multi-group .

The problem of influencing Ising systems on networks has recently been
considered in Refs. lynn2016maximizing ; lynn2017statistical ;
lynn2018maximizing ; hindes2017large . This work differs from much of the past
work in Influence Maximisation which considers non-equilibrium spreading and
cascade processes kempe2003maximizing . We believe the equilibrium modelling
paradigm is relevant when people’s beliefs evolve slowly, which may be the
case for vaccine sentiment colijn2017toward . The control of spin systems is
also relevant in quantum optimal control day2019glassy , spintronics
rotskoff2017geometric  and neuroscience lynn2019physics .

In this paper we provide the first analysis of the Ising influence problem
from the perspective of partially observed networks with modular structure. We
consider influencing Ising systems with strong ambient (or background) fields,
which, to our knowledge, have only been considered in Refs.
burioni2015enhancing  and godoy-lorite2020inference . Ising systems on modular
networks possess multiple ‘metastable’ solutions which correspond to different
blocks of spins being aligned or anti-aligned. In practice, we can estimate
the metastable state by surveying a subset of the nodes. We concentrate on
influence strategies which peturb the system from its current metastable state
by applying a low but sustained control field (see Section I.4). However, the
findings from this paper can likely be extended to other control paradigms.
These might include state-dependent feedback controls, parametric influence
strategies, where we can tune Ising model or network model parameters godoy-
lorite2020inference , or influence strategies which involve driving a switch
between different metastable solutions rather than perturbing them
hindes2017large .

Sections I.1, I.2 and I.3 introduce SBMs, the Ising model and the Ising
influence problem. Following this, we state our assumptions about the relevant
parameter regimes in the study of large scale social influence (Section I.4).
After this, we develop a mean-field based approach for influencing Ising
systems on SBMs. We demonstrate the effectiveness of this approach by
evaluating it on a two block SBM (Section II.1). In Section II.2 we explore
how optimal influence strategies are impacted by the temperature and
metastable state in the presence of ambient fields. This allows us to arrive
at heuristic models of how we can influence Ising systems with ambient fields
given only knowledge of the node embedding (or social kernel). In Section II.3
we demonstrate the performance of our influence strategies on data from a
large attributed social network. We show that we can influence the system
using a coarse graining which only requires node attribute knowledge (age,
location) rather than network microdata. We also show that point estimates of
the state can be used to influence Ising systems.

###  I.1 Stochastic block models

Consider an undirected graph consisting of $N$ nodes, each of which belongs to
one of $q$ non-intersecting subsets with fixed sizes: $N_{i}$ for
$i\in\\{1,...,q\\}$. Let $\underline{t}$ be the vector describing the block
membership of the individual nodes. That is, $t_{i}=k$ if node $i$ is in block
$k$. The connection probabilities of nodes in an SBM are determined by the
entries of a $q\times q$ affinity matrix, $\Omega$. The elements of the
symmetric adjacency matrix $A$ are Bernoulli trials with a connection
probability given by:

$\mathbb{P}(A_{ij}=A_{ji}=1)=\Omega_{t_{i},t_{j}}.$ (1)

The _expected adjacency matrix_ , $\mathbb{E}(A)$, will be an $N\times N$
block matrix with elements $\Omega_{t_{i},t_{j}}$.

Another matrix which we will use is the _coupling matrix_ , $\mathcal{K}$.
This is a $q\times q$ matrix for which element $\mathcal{K}_{xy}$ represents
the expected number of links from a randomly selected node in block $x$ to
nodes in block $y$. This matrix is asymmetric in general, however, it will be
symmetric when $N_{x}=N_{y}\quad\forall\quad x,y$. The entries of
$\mathcal{K}$ will be constrained such that for $x\neq y$:

$\mathcal{K}_{xy}=\frac{N_{y}}{N_{x}}\mathcal{K}_{yx}.$ (2)

The coupling matrix can be obtained from:

$\mathcal{K}=\Omega\mathcal{N},$ (3)

where $\mathcal{N}=\mathrm{diag}(N_{1},N_{2},...,N_{q})$ is a matrix
containing block sizes on the diagonal.

The expected total number of edges from block $x$ to block $y$ will be given
by:

$E_{xy}=\begin{cases}\frac{N_{x}(N_{x}-1)}{2}\Omega_{xy}\quad\mathrm{if}\quad
x=y\\\ N_{x}N_{y}\Omega_{xy}\quad\mathrm{if}\quad x\neq y\end{cases}$ (4)

Given network data at block level, this formula can be inverted in order to
obtain a point estimate of $\Omega_{ij}$.

###  I.2 Ising systems on complex networks

In the Ising model mackay2003information ; newman1999monte , each node $i$ is
assigned a spin, $s_{i}\in\\{-1,+1\\}$. In the context of social networks this
might represent the opinions of individuals about a political issue or an
opinion on a scientific topic such as climate change or vaccines. The spin
configuration is described by the vector
$\underline{s}=(s_{1},s_{2},...,s_{N})$. A particular individual’s spin is a
random variable which is dependent on that of its neighbours. The spins are
coupled via the graph with adjacency matrix $A$, which we will assume to be
undirected with no self loops.

In the Ising model the spins tend to align so that the energy is minimized.
The energy is given by:

$E(\underline{s})=-\frac{1}{2}\sum_{ij}A_{ij}s_{i}s_{j}-\sum_{i}g_{i}s_{i},$
(5)

where $g_{i}$ represents an external field acting on spin $i$. It will be
appropriate to decompose $\underline{g}$ into a component $\underline{h}$
which we can control and an ambient field $\underline{b}$ which we cannot
influence so that: $g_{i}=h_{i}+b_{i}$. The probability of the system being in
a particular state, $\underline{s}$ is given by the _Boltzmann distribution_ :

$\mathbb{P}(\underline{s})=\frac{1}{Z}e^{-\beta E(\underline{s})},$ (6)

where $\beta$ is the inverse temperature and $Z=\sum_{\underline{s}}e^{-\beta
E(\underline{s})}$ is the partition function. This tells us that systems are
exponentially less likely to be in states with higher energy. Evaluating this
expression analytically requires a sum over $2^{N}$ possible values of
$\underline{s}$. This is not feasible for moderately sized systems.
Consequently, evaluating the distribution over states typically requires Monte
Carlo simulations (see Supplementary Section 1.3) or analytic approximations.

The state of an Ising system can be summarised by the magnetisation:

$M=\frac{1}{N}\sum_{i=1}^{N}\langle s_{i}\rangle,$ (7)

where $\langle\cdot\rangle$ represents an ensemble average over the dynamics.
We will have $M=\pm 1$ if all of the $N$ spins are aligned in the positive or
negative directions.

###  I.3 The Ising influence maximisation problem

When influencing individuals’ beliefs the goal may not necessarily be to align
all individuals in a particular direction. Instead, we may want to maximise
the diversity of opinions individuals are exposed to matakos2018tell .
Depolarising the system in this way allows us to combat the formation of ‘echo
chambers’ which are known to form within the context of climate
williams2015network  and vaccine schmidt2018polarization  sentiment. In other
cases, there is strong motivation to promote a certain belief or behaviour. In
the context of public health, we might wish to promote hand washing practices
for the sake of improving health outcomes aboud2012challenges , mitigating
against the spread of disease through global transportation networks
nicolaides2019hand  and containing outbreaks within hospitals
mastrandrea2015enhancing .

In Section I.2 we introduced the magnetisation $M$ as a metric for the amount
of polarisation in an Ising system. In this study, we seek to find external
fields $\underline{h}$ which maximise $M$ given some constraints. In the
context of political votes, this would correspond to finding the optimal way
to distribute a budget in order to maximise the vote share for a particular
party.

Recent work in Refs. lynn2016maximizing ; lynn2017statistical ;
lynn2018maximizing  has considered the problem of maximising the magnetisation
of an Ising system using an external control field. They refer to this as the
_Ising Influence Maximisation_ (IIM) problem. In Ref. lynn2016maximizing  they
introduce the IIM and a mean-field based optimisation algorithm for solving
the problem. The approximation performs well apart from close to the critical
temperature $\beta_{c}$ (defined below). At high temperatures
($\frac{\beta}{\beta_{c}}\ll 1$) it is preferable to target high degree nodes,
while at low temperatures ($\frac{\beta}{\beta_{c}}\gg 1$) it is better to
spread the budget among low degree nodes. The mechanics of this problem are
explored further in Ref. lynn2017statistical  which considers influence
strategies in the limit of low field budgets.

In the Ising Influence Maximization problem, the Ising system is influenced
both by a field $\underline{h}=(h_{1},h_{2},...,h_{N})$ which we can control
as well as an ambient field $\underline{b}=(b_{1},b_{2},...,b_{N})$ which is
not controlled. We consider a constraint on $\underline{h}$ of the form:

$|\underline{h}|=\sum_{i=1}^{N}|h_{i}|\leq H,$ (8)

where $H$ is the total field budget. For this form of field budget we can tune
the influence on each node continuously. It is also possible to consider
different constraints, for example, that of identifying the optimal set of
‘target nodes’ to have their states pinned lynn2018maximizing . In Ref.
lynn2016maximizing  the _Ising Influence Maximisation_ problem is defined as
follows: given an Ising system $A$, $\underline{b}$, $\beta$ and a budget $H$,
find a feasible control field $\underline{h}$, s.t $|\underline{h}|\leq H$,
that maximises the magnetisation. That is, find an optimal control field
$\underline{h}^{*}$, such that:

$\underline{h}^{*}=\arg\max_{|\underline{h}|\leq
H}M(\underline{b}+\underline{h}).$ (9)

The value of $M$ for a particular $\underline{h}$ can be estimated using the
mean-field approximation. It is also possible to estimate it using higher
order variational approximations lynn2018maximizing ; opper2001advanced .

Previous work on this topic provides a methodology for controlling Ising
systems, however, it does not address the issue of scalability to large
networks. In this paper we provide two improvements to the methods presented
in previous work. Firstly, we present an efficient way to estimate the
magnetisation gradient $\nabla_{\underline{h}}M$ (also referred to as the
_susceptibility vector_). This vector, which has elements
$\big{(}\nabla_{\underline{h}}M\big{)}_{i}=\frac{\partial M}{\partial h_{i}}$,
quantifies the response of $M$ to small changes in $\underline{h}$ (see
Supplementary Section 1.2.1). Secondly, we present a coarse grained approach
for influencing Ising systems on modular networks (introduced below).

Under the mean-field approximation we can estimate the magnetisation of an
Ising system by solving a set of self-consistency equations
yedidia2001idiosyncratic ; mackay2003information :

$m_{i}=\tanh\bigg{(}\beta\big{(}\sum_{j=1}^{N}A_{ij}m_{j}+b_{i}+h_{i}\big{)}\bigg{)},$
(10)

for $i=1,...,N$. These equations can be evaluated numerically using fixed
point iteration (Algorithm 1 in Supplementary Section 1.1). We will denote the
average mean-field magnetisation by $M^{MF}=\frac{1}{N}\sum_{i=1}^{N}m_{i}$.

For small values of $\beta$ and $\underline{h}=\underline{b}=0$, Equation. 10
has a single solution at $\underline{m}=0$. For larger values of $\beta$ there
exist solutions with non-zero magnetisation. The point at which these non-zero
solutions become stable is associated with the critical (inverse) temperature
$\beta_{c}$, which we use to set the temperature scale. $\beta_{c}$ can be
identified by considering the conditions required for $\underline{m}=0$ to be
the only solution to Equation 10 (see Supplementary Section 1.4).

Given a fixed field budget $H$ we can estimate the optimal control field
$\underline{h}^{*}$ using projected gradient ascent. At each timestep $t$ of
this algorithm we compute $\nabla_{\underline{h}}M^{MF}$ and make a step in
the direction of positive gradient
$\underline{h}^{t}\rightarrow\underline{h}^{t}+\epsilon\nabla_{\underline{h}}M^{MF}$
before projecting the result down to the set of fields which satisfies the
budget constraint (see Supplementary Section 1.2.1 for details).

For systems with block structure we can approximate the magnetisation further.
In Ref. lowe2019multi-group  the authors show, for an Ising system on a two
block SBM, in the $N\rightarrow\infty$ limit, the average magnetisation of
each block converges to that of the fully connected model if the graph is
sufficiently dense and homophilous. We make the strong, but intuitive,
assumption that these results can be extended to the case of a general number
of blocks. This is not likely to be the case for all parameter regimes,
however, our results indicate that this form of mean-field approximation
performs well for a wide range of relevant parameter values.

For Ising systems on networks with block structure we can estimate the
magnetisation at block level by solving a coarse grained version of Equation
10 of the form:

$m_{B_{x}}=\tanh\bigg{(}\beta\big{(}\sum_{y=1}^{q}\mathcal{K}_{xy}m_{B_{y}}+\tilde{b}_{B_{x}}+\tilde{h}_{B_{x}}\big{)}\bigg{)},$
(11)

for block index $x=1,...,q$, where $m_{B_{x}}$, $\tilde{b}_{B_{x}}$ and
$\tilde{h}_{B_{x}}$ are, respectively, the magnetisation, ambient field and
control field applied to block $x$. We assume that all nodes in a given block
experience the same ambient field. The derivation of this equation is given in
Supplementary Section 1.2.2.

Ising systems with general connectivity structure and ambient fields can have
a complicated solution structure with many metastable states
aspelmeier2006free-energy . For opinion dynamics on weakly coupled modular
networks the different metastable states correspond to solutions where spins
within each block are mostly aligned in the positive or negative directions
suchecki2009bistable-monostable ; lambiotte2007coexistence ;
lambiotte2007majority . In this paper we consider interventions which aim to
maximise $M$ (taking the positive direction as a default) given the system is
in a particular metastable state. When simulating Ising systems, the system
will equilibrate to a metastable state, and then remain in it for an
exponentially long amount of time before transitioning to another due to
fluctuations hindes2017large . Consequently, this assumption amounts to
identifying practically relevant ‘moderate timescale influence strategies’,
which will differ from those which are optimal in the infinite time horizon
case.

###  I.4 Parameter regimes in large-scale public information campaigns

In this paper, we are motivated by influence problem in the context of large-
scale public information campaigns. This places constraints on the parameter
regimes which we will consider when deriving influence strategies:

  1. 1.

_Strong ambient social fields._ At the level of populations an individual’s
demographic attributes will be highly predictive of their social outcomes
burioni2015enhancing ; opoku2019parameter ; godoy-lorite2020inference . In the
context of Ising models, this corresponds to a dominating effect from ambient
fields (with weaker fields being sufficient in the low temperature setting).

  2. 2.

_Some access to node-state microdata._ We get full or partial survey
information about individual’s demographic attributes and node states (e.g.
voting preferences) and their connectivity. However, it will not be possible
to obtain a detailed view of an individual’s social network.

  3. 3.

_Limited computational budget._ Even if we can obtain network data at a large-
scale, we might be limited by the fact that it is computationally costly to
solve optimisation problems on large graphs.

  4. 4.

_Limited field budget._ It is costly to perform large-scale public information
campaigns. We will therefore consider values of $H$ which are sufficiently
small to induce changes in $M$ of only a few percent. Improvements of this
magnitude constitute substantial progress in social applications. Low but
sustained interventions under moderate timescales will result in mild
perturbations of the current metastable state.

When designing influence strategies we assume knowledge of either node or
block level coordinates, the ambient fields associated with these blocks
$\tilde{\underline{b}}$ or nodes $\underline{b}$, the inverse temperature
$\beta$ and coupling strengths between the nodes $A$ or blocks $\mathcal{K}$.
We also assume that we know the current metastable state of the system (in
practice this might be obtained from social surveys). The influence strategies
we consider are open-loop and time-independent as they do not rely on
information about the system state, apart from an initial estimate of the
current metastable state. Open-loop controls with pulsed fields might present
a significant improvement for a given average field budget.

We do not make any claims about the optimality of the influence strategies
considered. Our motivation is closer to that of Ref. kolumbus2017phase  which
shows that for the Maximum Influence Problem on the cascade model that, apart
from in a few special cases, simple heuristic strategies can perform
comparable to costly optimisations.

##  II Results

###  II.1 A scalable approach for influencing Ising models on networks with
block structure

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 1: The MF-IIM control derived using only the block structure of an SBM
can perform comparably to a control which uses the whole graph structure. (a)
Plot showing the SBM defined in text with nodes coloured according to the
value of the IIM control. Nodes in block 1 (LHS) have an average degree of
12.5 while those in block 2 (RHS) have a smaller average degree of 5. (b)
Histogram showing the distribution of $\underline{h}_{\mathrm{IIM}}$ values
for the two blocks for $\beta=\frac{3}{2}\beta_{c}$, $H=10$. The average
control applied to the second, lower, degree block is significantly smaller
than that applied to the first block. We evaluated the controls on the same
SBM network drawn from the ensemble with coupling matrix in Equation. 12 for a
range of $\beta$ and $H$ values. (c), (d), (e) Illustrate how $\Delta
M(\underline{h}_{\mathrm{IIM}})$ (blue circles) and $\Delta
M(\underline{h}_{\mathrm{block}})$ (red squares) behave as a function of $H$
for $\beta=0.5\beta_{c}$ (c), $1.2\beta_{c}$ (d) and $1.5\beta_{c}$ (e). (f)
and (g) show the behaviour of $M_{MC}(\underline{h}_{\mathrm{unif}})$ and
$\delta M_{F}^{Block}$ respectively for different values of $\beta$. Error
bars indicate the standard error on the mean from 15 Monte Carlo simulations.
Parameters used for the numerical simulations are given in Supplementary
Section 1.5.

In this section we compare the structure of the solutions obtained by the full
graph IIM and the block level approximation on a two block SBM. We then show
that the magnetisation obtained using the block level IIM can be comparable to
that obtained by using the full graph IIM on the same SBM.

We will consider an Ising system on the two block SBM with the coupling
matrix:

$\mathcal{K}=\begin{pmatrix}10&2.5\\\ 2.5&2.5\end{pmatrix}.$ (12)

This system consists of a strongly coupled block (block 1) coupled to a weakly
coupled one (block 2). We set the block sizes to be equal with
$N_{1}=N_{2}=250$. A draw from this ensemble is shown in Figure 1a.

We evaluate the performance of the different influence strategies using Monte
Carlo simulations and explore how the performance is impacted by the
parameters $H$ and $\beta$. Let $M_{MC}(\underline{h})$ be the average
magnetisation of the system evaluated using Monte Carlo simulations. Define
the _magnetisation markup_ for control h to be:

$\Delta
M_{MC}(\underline{h})=M_{MC}(\underline{h})-M_{MC}(\underline{h}_{\mathrm{unif}}),$
(13)

where $\underline{h}_{\mathrm{unif}}$ represents the uniform baseline strategy
for which $h_{\mathrm{unif},i}=\frac{H}{N}$.

We implement the projected gradient ascent algorithm described above at the
full graph (Algorithm 2) and at the level of blocks (Section 1.2.2) for a
range of values of $H$. We consider 3 different values of
$\beta_{c}=\frac{1}{\lambda_{1}}$ where $\lambda_{1}$ denotes the largest
eigenvalue of the adjacency matrix, $A$, of the sampled SBM. The parameters
used in the projected gradient ascent algorithm are provided in Supplementary
Section 1.5.

The average magnetisations for controls were estimated by running Markov-Chain
Monte Carlo with the Metropolis dynamics described in Section 1.3. Simulations
are initialised with a random initial condition ($M\approx 0$). For each value
of the field budget we determine the magnetisation by taking the average value
obtained from 15 simulations with a burn-in time of $T_{\mathrm{burn}}=2\times
10^{4}$ and run time of $T=10^{4}$ steps.

Figures 1c,1d and 1e show $\Delta M_{MC}(\underline{h})$ as a function of $H$
for $\beta=0.5\beta_{c},1.2\beta_{c}$ and $1.5\beta_{c}$ respectively. $\Delta
M_{MC}(\underline{h}_{\mathrm{IIM}})$ lies close to $\Delta
M_{MC}(\underline{h}_{\mathrm{Block}})$ for a range of values of $H$ and
$\beta$. This indicates the presence of scenarios in which the block level IIM
can be used to effectively Ising influence Ising systems on SBMs. We would
expect $\Delta M_{MC}(\underline{h}_{\mathrm{Block}})$ to be much smaller in
the case where the two blocks have similar average degrees.

We also compute the quantity $\delta
M_{F}^{Block}=\frac{M_{MC}(\underline{h}_{\mathrm{block}})}{M_{MC}(\underline{h}_{\mathrm{unif}})}$
(Figure 1g) which indicates the fractional increase in the magnetisation
relative to the uniform baseline. $\delta M_{F}^{Block}$ is largest for
$\beta=0.5$ for which we see a relative increase in the magnetisation of at
least $20\%$. For larger $\beta$ we observe a smaller increase of up to $\sim
5\%$. The larger value of $\delta M_{F}^{Block}$ in the former case coincides
with the case when the $M_{MC}(\underline{h}_{\mathrm{unif}})$ is small
(Figure 1f).

###  II.2 Ising influence in the presence of ambient fields

In this section we explore the influence strategies which arise in Ising
systems with ambient fields. We first consider the Ising influence problem on
a simple model of a polarised society. The insights drawn from this analysis
provide us with some more informed baseline control strategies to be applied
to Ising systems with ambient fields.

We will consider an SBM consisting of 3 blocks arranged in a chain with the
coupling matrix:

$\mathcal{K}=\begin{pmatrix}10&2.5&0\\\ 2.5&7.5&2.5\\\ 0&2.5&10\end{pmatrix},$
(14)

with block sizes of: $N_{1}=N_{2}=N_{3}=400$. Let
$\underline{B}_{\mathrm{block}}$ be the vector of ambient fields acting at the
level of blocks. We will take this to be of the form:

$\underline{B}_{\mathrm{block}}=(+B,0,-B),$ (15)

where the parameter $B$ controls the magnitude of the field applied. This
model consists of two communities with opposing ‘external bias’ which can only
communicate via an intermediate group with no external bias. The coupling
structure here shares similarity with that between three different age groups
obtained by fitting an Ising model to health screening data in Ref.
burioni2015enhancing  and is consonant with the geometric kernel inferred in
godoy-lorite2020inference . Figure 2a shows a typical draw of an SBM from this
system. The average degrees of the nodes in the SBM with the coupling matrix
in Equation 14 will be equal. This homogeneity in the node degrees allows us
to focus on the effects of the ambient field on the optimal influence
strategy.

Before studying the influence problem on this network, we first enumerate the
values which $M$ can take for different $\beta$ values in the absence of a
control field. We compute the phase diagram of the system using two different
mean-field approximations (full and block level) and Monte Carlo simulations.
We identify different metastable states by initialising the fixed point
iterations and Monte Carlo simulations at different random initial conditions
(see methods in Appendix 3).

The phase diagram for the three block system is shown in Figure 2b. The phase
diagram consists of three different regimes containing one, two and four
metastable states. The results shown in Figure 2b demonstrate that the block
level mean-field approximation is able to reproduce the shape of the phase
diagram obtained using Monte Carlo simulations and the full graph mean-field
approximation. The main discrepancies between the different methods occur
close to the two transition points.

(a)

(b)

(c)

(d)

(e)

Figure 2: The optimal low $H$ block level IIM strategy depends on $\beta$ and
the metastable state. (a) SBM drawn from the ensemble with the coupling matrix
specified in Equation 14 for $N=1200$. The ambient fields experienced by the
nodes are indicated on the colour bar. (b) Plot showing the different
solutions for the magnetisation of the three-block system as a function of
$\beta$. The plot shows the different solutions identified using the block
level mean-field approximation (black circles) and the full graph mean-field
approximation (blue squares). These are overlaid onto a heatmap showing the
probability distribution of magnetisation values obtained using Monte Carlo
simulations. For each value of $\beta$ we sampled from two Monte Carlo chains
with a burn in time of $2\times 10^{4}$ and run time of $10^{4}$ steps for
each of 25 random initial conditions (see Supplementary Section 3). The
histograms (for a given $\beta$) are normalised by the maximum count so that
the different metastable states are easier to identify. The value of
$\beta_{c}$ was computed from the spectral radius of the SBM. (c), (d) and (e)
Show the behaviour of the magnetisation gradient
$\nabla_{\underline{h}}M^{MF}|_{\underline{h}=0}$ of each of the three blocks
as a function of $\beta$. Solutions corresponding to the different metastable
states are obtained by initialising Algorithm 1 with different uniform random
initial conditions in $[-1,1]^{3}$.
$\nabla_{\underline{h}}M^{MF}|_{\underline{h}=0}$ is computed by numerically
solving Equation 27 in each instance. Markers are coloured and styled
according to the average magnetisation in order to reveal which metastable
state the values of the susceptibility are associated with. For large $\beta$,
the values of the susceptibilities obtained for different metastable solutions
overlap with each other. However, in all three cases it is possible to
determine which of the three blocks has the largest susceptibility.

####  II.2.1 Low $H$ solutions to the IIM problem with ambient fields

We can characterise the low field budget solutions to the IIM problem by
computing the susceptibility vector
$\nabla_{\underline{h}}M^{MF}|_{\underline{h}=0}$ lynn2017statistical . We
performed this computation at the level of blocks for a range of different
values of $\beta$. Different metastable states were identified using random
initial conditions (see Supplementary Section 3).

The second row of Figure 2 shows each of the components of
$\nabla_{\underline{h}}M^{MF}|_{\underline{h}=0}$ as a function of $\beta$.
The scatter points are coloured according to the different magnetisation
values in order to reveal the different metastable states identified in Figure
2b. The susceptibility was found to depend on the metastable state that the
system is assumed to be in.

In the noisy or high temperate (low $\beta$) setting, the susceptibility of
all three blocks approaches the same value. This suggests it is optimal to
spread the budget uniformly, which is intuitive as the blocks have the same
average degree. For large $\beta$, the optimal low $H$ block level influence
strategy is to focus all of the budget on one of the three blocks. This is
block 1 for the most negatively aligned metastable state, block 2 for the two
metastable states with moderate magnetisation and block 3 for the metastable
state with positive alignment. Based on these findings we suggest three
different heuristic influence strategies which do not require any knowledge of
the SBM connectivity structure.

  1. 1.

Swing voter. When the component of
$\nabla_{\underline{h}}M^{MF}|_{\underline{h}=0}$ associated with block 2 is
largest it is optimal to spend most of the budget on the block with no ambient
external field. We will refer to this as the _swing voter_ strategy as it
mirrors the tactic used in political campaigns of targeting sub populations
who have no strong bias either way.

In general, it is not reasonable to expect nodes to have ambient fields
exactly equal to zero. Consequently, we will consider the set of nodes for
which:

$SV(\psi)=\\{i:|b_{i}|<\psi\\},$ (16)

where $\psi$ is a positive constant. We will use a heuristic which spreads
influence equally among the individuals in this set. We define the _swing
voter_ influence strategy, $\underline{h}_{SV}(\psi)$, to be:

$h_{SV,i}=\begin{cases}\frac{H}{|SV(\psi)|}\quad\mathrm{if}\quad i\in
SV(\psi)\\\ 0\quad\mathrm{otherwise}.\end{cases}$ (17)

  2. 2.

Negative canceller. For large $\beta$ there is a metastable state in which all
of the spins are aligned in the positive direction. In this state the
component of $\nabla_{\underline{h}}M^{MF}|_{\underline{h}=0}$ associated with
block 3 is largest, which suggests that all of the budget should be applied to
the block with negative ambient field. The corresponding heuristic strategy we
consider involves spending budget to cancel out negative ambient fields. Let

$\underline{h}_{NC}=\frac{H}{|\underline{h}^{-}|}\underline{h}^{-},$ (18)

where $\underline{h}^{-}$ is the vector with elements:

$h^{-}_{i}=\begin{cases}-b_{i}\quad\mathrm{if}\quad b_{i}<0\\\
0\quad\mathrm{otherwise}\end{cases}.$ (19)

  3. 3.

Survey-snapshot influence strategy. As noted in Section I.4 it is common to
expect that we might get information about the system state. Consequently, for
our final null model, we consider the case where only point estimates of the
magnetisation of the blocks are available.

In analogy to a social survey taken at a particular point in time, we will use
a sample of node states to estimate the magnetisation. We estimate the
magnetisation of each block $\hat{M}_{B_{x}}(F_{\mathrm{samp}})$ at a
particular point in time by sampling the states of $\lceil
F_{\mathrm{samp}}N_{B_{x}}\rceil$ nodes without replacement from each block
$x$. We perform this sampling after waiting for the Monte Carlo chain to burn
in. We then approximate the susceptibility of a particular block using the
mean-field approximation. Differentiating Equation 10 with respect to $h_{i}$
it is possible to show that the mean-field susceptibility of a node to its own
field is equivalent to $\beta(1-m_{i}^{2})$. Thus, the vector with elements
$(1-\hat{M}_{B_{x}}^{2})$ can be used to approximate the susceptibility of
each block. We will refer to this as the _survey-snapshot_ influence strategy.
A more robust estimate of the susceptibility of the system would require
multiple observations (which can be used to compute the covariance matrix) or
knowledge of the connectivity between the blocks (which we assume is not
available in the case of this null model).

We summarise the information required by each of the strategies in
Supplementary Table S1.

###  II.3 Influencing an Ising system on a large homophilous social network

In this section we validate the methodology further by applying the influence
strategies to data obtained from an online social network. We consider a
significantly larger network than previous ($N=29,582$) in order to
demonstrate scalability of the influence strategies. We consider an attributed
network dataset obtained from the online social network Pokec takac2012data
(see Supplementary Section 4.1 for details). We consider a subset of nodes
from the full network in a set of regions in the vicinity of Bratislava. We
coarse-grain the resulting network into twelve blocks based on the location
and age of individuals (see Supplementary Section 4.2). We find strong
homophily in both age and region (Figure 3a), indicating that this form of
coarse graining provides an appropriate summary of the network structure.

The Pokec social network dataset does not encode any ambient fields associated
with the nodes. Consequently, we will impose an artificial ambient field to
replicate what we might expect in a real system. Vaccine sentiment is known to
covary with age in some countries de_figueiredo2016forecasted . Motivated by
this, we will consider an ambient field which varies linearly with age (see
Supplementary Section 4.3) with gradient $g$. The system considered has
numerous metastable states in the low temperature regime (See Supplementary
Figure S2). We will restrict our analysis to the most negative metastable
state which can be obtained by initialising the fixed-point iteration and MC
simulations at $(-1,-1,...,-1)$. In the analysis below we set
$\beta=8\beta_{c}$. This choice of $\beta$ corresponds to a ‘moderately’ low
temperature in which spins are broadly aligned with their blocks but still
allows for the possibility for spins to flip due to being influenced or
thermal fluctuations (See Supplementary Figure S2).

We derive low budget influence strategies by computing
$\nabla_{\underline{h}}M^{MF}$ and $\nabla_{\underline{h}_{B}}M^{MF}$ (see
Equations 27 and 41 in the Supplement). Normalising and multiplying by $H$ in
each case allows us to obtain a control with budget $H$. For the linear
ambient field considered, the 22-28 age group has the lowest magnitude ambient
field. We select this set of nodes to be $SV(\psi)$ when deriving the swing
voter based influence strategy. The negative canceller influence strategy was
computed from Equation 18. We generate a single instance of the survey-
snapshot control for each value of $g$. We select $F_{\mathrm{samp}}=1$ in
order to obtain an upper bound on the performance of the survey-snapshot
control. The impact of using smaller sub-samples is explored in Supplementary
Section 4.5. For our simulations we have selected $H=2500$. This value of $H$
is sufficient to observe changes in magnetisation of a few percent but
sufficiently small to align with the assumption of low field budget (see
Section I.4).

Figure 3b shows how $\Delta M^{MC}$ varies as a function of the gradient of
the linear field $g$. In all cases, the control which uses knowledge of the
full graph performs best. Naturally, for low $g$, the block level mean-field
control performs best out of the coarse grained controls. For larger values of
$g$ the swing-voter control performs comparably to the block level influence
strategy. Consequently, in the high field case, knowledge of the ambient field
applied to nodes may be sufficient to influence the system effectively.

We refer to a strategy as being consistent if $\Delta M^{MC}$ is large for a
range of parameters. In the absence of an ability to pre-run simulations it
makes sense to choose a consistent strategy. For the parameters considered,
the block level influence strategy has the most consistent good performance.
If coarse grained network information is not available, the next most
consistent solution, would be the survey-snapshot strategy. The swing voter
heuristic can perform better than the survey-snapshot strategy in the presence
of strong ambient fields. However, the results in Figure 2 suggest that the
performance of the swing-voter and negative canceller strategies are dependent
on the particular metastable solution and are therefore less consistent. The
swing voter strategy also has a free parameter $\psi$.

(a)

(b)

Figure 3: Knowledge of coarse grained connectivity, ambient fields or point
estimates of the magnetisation can be used to influence an Ising system on an
online social network with ambient ambient fields. (a) Plot showing the
elements of the the coupling matrix between different demographic blocks in
the Pokec social network. The coupling matrix $\mathcal{K}$ was determined by
counting the number of edges between the different blocks and inverting
equations 4 and 3. The large diagonal elements indicate strong homophily
within blocks of the same age and region, while those on the off diagonals
indicate age homophily between blocks. (b) Plot showing $\Delta M_{MC}$ for
different influence strategies as a function of the linear field gradient $g$.
Shown for full graph, block level, swing voter, negative canceller and survey-
snapshot influence strategies for $H=2500$, $\beta=8\beta_{c}$, with
$\beta_{c}$ being the inverse of the spectral radius of the full graph.
Average magnetisation values were estimated by averaging over 15 Monte Carlo
chains with a burn in time of $3\times 10^{5}$ steps and a run time of
$10^{4}$ steps. Error bars indicate the standard error on the mean.

##  III Discussion

Ising models and their extensions are used to model phenomena including
opinion dynamics castellano2009statistical ; acemoglu2011opinion , cellular
signalling weber2016cellular  and the dynamics of neurons lynn2019physics  and
ecological systems nareddy2020dynamical . They also are an example of a Markov
Random field kindermann1980markov ; possolo1986estimation , which are used in
spatial statistics cressie1992statistics ; besag1974spatial  and Bolztmann
machine learning tanaka1998mean-field . Quantifying the response of Ising
models to ambient fields allows us to devise public health campaigns godoy-
lorite2020inference ; burioni2015enhancing , control the state of quantum
systems day2019glassy ; rotskoff2017geometric  and potentially devise targeted
therapies for mental illness lynn2019physics .

Many real-world networks are inherently modular fortunato2010community . We
have shown that we can use this modular structure to estimate $M$ and devise
computationally efficient influence strategies which do not require network
microdata. Our coarse-grained mean-field approach provides a scalable method
to effectively influence Ising systems in the low field budget, strong field
regime described in Section I.4. This methodology can be applied to large
networks where it is computationally expensive to evaluate
$\nabla_{\underline{h}}M^{MF}$ at the level of individual nodes. We also found
that heuristic strategies, which rely on knowledge of the state or ambient
fields, can effectively influence Ising systems. These strategies can be
applied in circumstances where it is not possible to infer connectivity
information.

Our simulations suggest that the approximation for the magnetisation of two-
block SBMs presented in Ref. lowe2019multi-group  generalises to the case of
multiple blocks. We have also provided evidence that MSCW models
fedele2013inverse ; fedele2017inverse ; opoku2018multipopulation ;
gallo2009parameter ; gallo2009equilibrium ; opoku2019parameter  can be
extended to the case when there is randomness in the graph structure. We
expect that our results generalise to other models of opinion dynamics and
spatially embedded networks.

The problem of influencing dynamical systems defined on unobserved or noisy
networks has only been considered in a handful of studies wilder2017uncharted
; wilder2017influence ; erkol2018influence . Refs. wilder2017uncharted ;
wilder2017influence ; erkol2018influence  consider influencing the dynamics of
the independent-cascade model. This deterministic modelling framework fails to
capture the impact of noise and does not allow us to model the formation of
opinions due to many interactions occurring over a long period of time
lynn2016maximizing ; lynn2017statistical . The influence strategies considered
in Refs. wilder2017uncharted ; wilder2017influence  involve querying the state
of individual nodes in the graph, while Ref. erkol2018influence  explicitly
models the errors in the network structure. The opinion dynamics model and
influence strategies used in Refs. wilder2017uncharted ; wilder2017influence ;
erkol2018influence  therefore provide differing but complementary frameworks
to the one presented in this paper.

In this study we consider the case where partial network information comes in
the form of discrete or continuous node coordinates and associated connection
probabilities. In the continuous case, connection probability information has
been used to forecast dynamics on spatial networks via diffusion
fisher2017social ; smith2018using  or PDE based approaches
hoffmann2018partially-observed ; lang2017random . Our work demonstrates that
it is also possible to influence dynamics on networks using this level of
information. Future work will be required to generalise our results to
spatially embedded networks. Given a node distribution and connectivity
kernel, the predictability of dynamical properties in spatial networks is
impacted by dimensionality and inhomogeneity in the node distribution
garrod2018large . Consequently, we expect our ability to influence dynamics on
spatial networks to depend on the same factors.

Ising systems on modular networks possess a complicated solution structure
with multiple metastable states. Our findings highlight the importance of
understanding this structure when deriving influence strategies. The solutions
to Equation 10 and their cardinality are controlled by the temperature
parameter $\beta$. Our findings compliment existing results on the importance
of $\beta$ in influencing Ising systems lynn2017statistical  and highlight the
importance of having robust methods for estimating it fedele2017inverse ;
godoy-lorite2020inference .

In Section II.3 we selected node groups based on attributes. A more informed
partition of the graph could be obtained using a community detection
algorithm. Typically, this would require network information, however, it has
been shown that community detection can be performed based on other forms of
information, such as time series associated with the nodes
hoffmann2020community ; schaub2019blind ; peixoto2019network . We might also
include more detailed information about the node degrees into influence
strategies using using degree corrected SBMs karrer2011stochastic  or variants
of the mean-field approximation suchecki2009bistable-monostable .

In this paper we consider a weak but sustained field applied to the system. A
complementary paradigm involves switching between metastable states rather
than perturbing them hindes2017large . This might be achieved using impulses
of control fields, comparable to the ‘bang-bang’ control protocols considered
in day2019glassy . These approaches may allow us to achieve a significant
increase in magnetisation with a relatively small time-integrated control
field budget.

Our work acts as a proof-of-concept to demonstrate that partial network data
from social surveys may be sufficient for network based interventions. Recent
work in Ref. mcpherson2019network  applies social connectivity parameters
inferred from one survey to a larger dataset to model large scale social
connectivity. They use this model to predict how people’s attitudes arise from
a combination of network contacts and external covariates. In this paper we
have demonstrated that it might be possible to go one step further and
influence social outcomes given this information.

Non-uniform Ising influence strategies can be effective when nodes have
heterogeneous degrees (Figure 1) or experience different ambient fields
(Figures 2 and 3). We found that the majority of influence strategies
considered perform well in the strong ambient field limit, which we argue is
the sociologically relevant setting to consider Ising systems in. This
indicates the possibility of effective public information campaigns which use
tools from network science whilst nonetheless having incomplete network data.

## Acknowledgements

This work has been supported by EPSRC grants EP/L016613/1 and EP/N014529/1.
The authors would like to thank Chris Lynn, Sarab Sethi, Antonia Godoy-Lorite,
Till Hoffman and Sahil Loomba for useful advice and comments on the
manuscript.

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Influencing dynamics on social networks without knowledge of network
microstructure
Supplementary material
Matthew Garrod, Nick S. Jones

##  1 Algorithms for mean-field Ising systems

###  1.1 Iterative schemes for solving the mean-field equations

Equations 10 cannot be evaluated analytically, however, they can be solved
using an iterative method. We will consider a fixed point iteration in which
we update the magnetisation of each node according to aspelmeier2006free-
energy ; lynn2018maximizing :

$m_{i}^{t+1}=(1-\gamma)m_{i}^{t}+\gamma\tanh\bigg{(}\beta\big{(}\sum_{j=1}^{N}A_{ij}m_{j}^{t}+g_{i}\big{)}\bigg{)}$
(20)

where we update the values in the order of their index (e.g. starting at $i=0$
finishing at $i=N$) (According to aspelmeier2006free-energy  the order in
which we update for the different spins does not matter). This iterative
method is sometimes referred to as _damped fixed-point iteration_
dahmen2016correlated ; zou2011improved . The damped fixed point iteration
scheme is summarised in Algorithm 1.

Input: Adjacency matrix $A$, inverse temperature $\beta$, external field
vector $\underline{g}$, tolerance $\tau$, damping term $\gamma$, initial
magnetisation $\underline{m}^{0}$.

Result: Approximate solution to the mean-field self-consistency Equations 10
$\underline{m}^{*}$.

Initialisation: $t\leftarrow 0$, $i\leftarrow 0$ ;

while  _$|\underline{m}^{t+1}-\underline{m}^{t}| >\tau$_ do

while  _$i <N$_ do

$m_{i}^{t+1}\leftarrow(1-\gamma)m_{i}^{t}+\gamma\tanh\bigg{(}\beta\big{(}\sum_{j=1}^{N}A_{ij}m_{j}^{t}+g_{i}\big{)}\bigg{)}$
;

$i++$ ;

end while

$t++$ ; $i\leftarrow 0$

end while

Return $\underline{m}^{t+1}$

Algorithm 1 Damped fixed point iteration

When solving the standard mean-field equations, standard fixed point (Picard)
iteration berinde2007iterative  with $\gamma=1.0$ is sufficient in order
secure convergence of the scheme aspelmeier2006free-energy . However, when
solving the self-consistency equations for higher order variational
approximations, such as the TAP approximation, it becomes necessary to use the
damped fixed point (Krasnoselskij) iteration in order to secure convergence
aspelmeier2006free-energy ; bolthausen2014iterative . Damped fixed point
iteration allows us to obtain the fixed points of maps which satisfy less
assumptions concerning their contractive properties (see theorem 3.2 in
berinde2007iterative ).

For general Ising systems there will not be a unique solution to 10. The
solution that we obtain using Algorithm 1 will depend on the state that we
initialise the system in. Apart from when otherwise specified, we will focus
on identifying the solution with the largest positive value of $M^{MF}$. For
general $A$ and $\underline{g}$ the basins of convergence for the dynamical
system defined in 20 will be difficult to identify. However, in practice we
find that initialising $\underline{m}^{0}=(1,1,...,1)$ results in convergence
to the solution with largest positive $M^{MF}$.

###  1.2 Control of Ising systems

####  1.2.1 Projected gradient ascent algorithm for Ising influence
maximisation

We will estimate $\underline{h}^{*}$ by using a gradient ascent based
approach. This standard approach for solving optimisation problems involves
initialising the system at some initial state $\underline{h}^{0}$, and then
making a series of steps in the direction of positive gradient of the
objective function. In this case our aim is to maximise the magnetisation $M$.
An accurate estimate of the gradient in the magnetisation could be obtained
using Monte Carlo simulations, however, this would be costly as a new estimate
of the gradient has to be made at each step. Consequently, we use the mean-
field approximation to approximate the gradient.

The magnetisation gradient and susceptibility. The gradient in the
magnetisation with respect to the control field $\underline{h}$ is defined by:

$\nabla_{\underline{h}}M=\bigg{(}\frac{\partial M}{\partial
h_{1}},\frac{\partial M}{\partial h_{2}},...,\frac{\partial M}{\partial
h_{N}}\bigg{)}.$ (21)

The elements of this vector can be written in terms of the _susceptibility
matrix_ $\chi$, which has elements:

$\chi_{ij}=\frac{\partial m_{i}}{\partial h_{j}}.$ (22)

Element $\chi_{ij}$ quantifies how changing the field experienced by node $j$
affects the magnetisation of node $i$. We will have:

$\frac{\partial M}{\partial h_{j}}=\frac{\partial}{\partial
h_{j}}\sum_{i=1}^{N}m_{i}=\sum_{i=1}^{N}\chi_{ij}.$ (23)

Under the mean-field approximation the susceptibility can be obtained by
differentiating Equation 10 in order to obtain:

$\chi^{MF}=\beta(I-\beta DA)^{-1}D,$ (24)

where $D$ is an $N\times N$ matrix with elements
$D_{ij}=(1-m_{i}^{2})\delta_{ij}$. In Ref. lynn2016maximizing  $\chi^{MF}$ is
obtained from Equation 24. This involves the inversion of an $N\times N$
matrix which becomes costly as the size of the graph increases.

In Ref. tanaka1998mean-field  they show how, under the mean-field
approximation the inverse of the susceptibility matrix is given by the matrix
with elements:

$S_{ij}=(\chi^{-1})_{ij}=\frac{1}{\beta}\frac{1}{1-m_{i}^{2}}\delta_{ij}-A_{ij}.$
(25)

The inverse of this matrix provides an estimate of the correlations under
linear response theory. In order to compute the gradient of the mean-field
magnetisation we only need the elements of $\nabla_{\underline{h}}M^{MF}$
rather than all of the elements of the susceptibility matrix. The gradient can
be written as:

$\nabla_{\underline{h}}M^{MF}=\chi^{T}\mathbf{1},$ (26)

where $\mathbf{1}=(1,1,...,1)$ is a vector of ones. Inverting and noting to
swap the transpose and inverse we obtain:

$S^{T}\nabla_{\underline{h}}M^{MF}=\mathbf{1}.$ (27)

This linear equation can be solved using standard linear algebra routines
which are more efficient than matrix inversion.

Given the gradient of the magnetisation at timestep $t$, we then make a step
of magnitude: $\epsilon\nabla_{\underline{h}}M^{MF}|_{\underline{h}^{t}}$
where $\epsilon$ is the step size parameter. Note that we normalise the
gradient by its magnitude. The size of the step which we take could vary with
iteration. Various algorithms including ‘line search’ based approaches exist
for estimating the optimum step size to take at each iteration during
optimisation procedures nocedal2006numerical . For simplicity we will consider
the case of fixed step size. We find that we can obtain convergence in a
feasible number of iterations with fixed step size in practice for the
parameters considered in this paper.

Satisfying the budget constraint.  If the proposed $\underline{h}^{t+1}$ does
not satisfy the imposed budget constraint (Equation 8) we identify a
projection of this vector which does. A naive implementation of this step can
be very slow. Consequently, it is of interest to find more efficient
algorithms for carrying out this projection. In order to improve the
performance we use an implementation of the methods discussed in Ref.
blondel2014large-scale . The code used for this step is available at:
https://gist.github.com/mblondel/6f3b7aaad90606b98f71. They discuss how
projections onto the $l_{1}$-ball can be treated using methods for performing
projections onto the simplex. These methods were found to be relatively
efficient compared to a naive implementation.

Description of the algorithm.  Combining the steps described above we obtain
an update rule for the control field:

$\underline{h}^{t+1}=\mathcal{P}_{|\underline{h}|\leq
H}\bigg{[}\underline{h}^{t}+\epsilon\nabla_{\underline{h}}M^{MF}\bigg{]},$
(28)

where $\mathcal{P}_{|\underline{h}|\leq H}\big{[}\cdot\big{]}$ represents the
projection into the set of fields which satisfy the budget constraint. We can
repeatedly update according to this rule until the change in the magnetisation
is less than some threshold parameter $a$. The resulting procedure for
determining the control field is shown in Algorithm 2. At each step of
Algorithm 2 we use Algorithm 1 to compute
$M^{MF}(\underline{b}+\underline{h}^{t})$.

Input: Adjacency matrix $A$, ambient field $\underline{b}$, step size
$\epsilon$, tolerance $a$, inverse temperature $\beta$, initial control field
$\underline{h}^{0}$.

Result: Estimate of the optimal control field $\underline{h}^{*}$.

Initialisation $t\leftarrow 0$ ;

while
_$|M^{MF}(\underline{b}+\underline{h}^{t})-M^{MF}(\underline{b}+\underline{h}^{t-1})|
>a$_ do

$\underline{h}^{t+1}=\mathcal{P}_{|\underline{h}|\leq
H}\bigg{[}\underline{h}^{t}+\epsilon\nabla_{\underline{h}}M^{MF}\bigg{]}$;

$t++$;

end while

Return $\underline{h}^{t+1}$

Algorithm 2 Projected gradient ascent

This algorithm is the same as that in lynn2016maximizing . However, we found
that computing the susceptibility from Equation 27, rather than Equation 24
resulted in a significant speed up. This, combined with the efficient method
used for computing the projection, was found to result in a significant
overall speed up. Numerical simulations suggest that the improvement can be
several orders of magnitude. This speed up meant that we can relatively
quickly obtain optimal fields for graphs containing a few thousand nodes. For
much larger graphs this algorithm will still scale poorly. As a result, we now
move on to describe methods which make use of the modular structure which
might be inherent in the system.

####  1.2.2 Block level approximation for the susceptibility

We now demonstrate how, under assumptions of block structure, we can reduce
the system of $N$ coupled equations for the mean-field magnetisation down to
$q$ equations at the level of blocks. After averaging over the disorder in the
block structure the treatment here follows a similar line of that of the
multi-species Curie-Weiss model (e.g. see Refs. fedele2017inverse ;
opoku2019parameter ). Recall that the mean-field self-consistency equations
are given by:

$m_{i}=\tanh\bigg{(}\beta\big{(}\sum_{j=1}^{N}A_{ij}m_{j}+g_{i}\big{)}\bigg{)}.$
(29)

We will assume that the adjacency matrix is drawn from a network with an SBM
structure as in Section I.1 with $q$ blocks. We define the average
magnetisation of block $k$ to be:

$m_{B_{k}}=\frac{1}{N_{k}}\sum_{k\in\mathcal{C}_{k}}m_{k},$ (30)

where, $\mathcal{C}_{k}$ is the set of nodes in block $k$.

We now make the assumption that we can average over the disorder in the
network structure by replacing the adjacency matrix with the expected
adjacency matrix (see Ref. lowe2019multi-group ). In this case, the mean-field
self-consistency equations become:

$m_{i}=\tanh\bigg{(}\beta\big{(}\sum_{j=1}^{N}\mathbb{E}(A)_{ij}m_{j}+g_{i}\big{)}\bigg{)}.$
(31)

In this system of equations the couplings experienced by nodes in the same
block will be identical. We further assume that all nodes in the same block
$k$ will experience the same total external field $\tilde{g}_{B_{k}}$. The
total field applied to the system can be characterised by the vector of length
$q$: $\tilde{\underline{g}}=(\tilde{g}_{B_{1}},...,\tilde{g}_{B_{q}})$. This
vector can be projected onto the full system of size $N$ by multiplying by the
block membership matrix to obtain:

$\underline{g}_{B}=G\tilde{\underline{g}},$ (32)

where $G$ is the block membership matrix for which $G_{ij}=1$ if node $i$ is
in block $j$ and $G_{ij}=0$ otherwise. The block level total external field
can be decomposed into the block level ambient field $\tilde{\underline{b}}$
and the block level control field $\tilde{\underline{h}}$ so that
$\tilde{\underline{g}}=\tilde{\underline{b}}+\tilde{\underline{h}}$. When
performing the optimisation at the level of blocks the budget constraint can
be written as:

$\sum_{k=1}^{q}N_{k}\tilde{h}_{B_{k}}=H.$ (33)

Under these assumptions the magnetisations of all nodes in block $k$ will be
equal to $m_{B_{k}}$. For a node $i$ in block $k$ the first term in the
brackets in Equation 31 can be written as:

$\sum_{j=1}^{N}\mathbb{E}(A)_{ij}m_{j}=(N_{k}-1)\Omega_{kk}m_{B_{k}}+\sum_{j\neq
k}N_{j}\Omega_{kj}m_{B_{j}}\\\ \approx\sum_{j=1}^{q}N_{j}\Omega_{kj}m_{B_{j}}$
(34)

Using the definition of the coupling matrix (Equation 3) we can write the
original set of $N$ coupled equations for the magnetisations of the nodes as a
set of $q$ equations describing the average magnetisations at the level of
blocks:

$m_{B_{x}}=\tanh\bigg{(}\beta\big{(}\sum_{y=1}^{q}\mathcal{K}_{xy}m_{B_{y}}+\tilde{g}_{B_{x}}\big{)}\bigg{)},$
(35)

for $x=1,...,q$. The _total magnetisation_ , which we wish to maximise, can be
written as:

$M^{MF}=\sum_{x=1}^{q}N_{x}m_{B_{x}}.$ (36)

The gradient in the magnetisation with respect to the block level control will
have elements:

$\big{(}\nabla_{\tilde{\underline{h}}_{B}}M^{MF}\big{)}_{y}=\frac{\partial}{\partial\tilde{h}_{B_{y}}}M^{MF}=\sum_{w=1}^{q}N_{w}\tilde{\chi}_{wy}^{MF},$
(37)

where $\tilde{\chi}^{MF}$ is the $q\times q$ block level mean-field
susceptibility matrix. This matrix will have elements:

$\tilde{\chi}_{xy}^{MF}=\frac{\partial
m_{B_{x}}}{\partial\tilde{h}_{B_{y}}}=\beta(1-m_{B_{x}}^{2})\bigg{(}\sum_{p=1}^{q}N_{p}\Omega_{xp}\frac{\partial
m_{B_{p}}}{\partial\tilde{h}_{B_{y}}}+\delta_{xy}\bigg{)}$ (38)

Defining $D_{M}$ such that $D_{M_{xy}}=(1-m_{B_{x}}^{2})\delta_{xy}$ allows us
to write an expression for the block level susceptibility matrix:

$\tilde{\chi}^{MF}=\beta D_{M}(I+\mathcal{N}\Omega\tilde{\chi}^{MF})$ (39)

Consequently, the analogue of the stability matrix (Equation 25) for the block
level system $\tilde{S}$ will therefore have the elements:

$\tilde{S}_{xy}=\big{(}\tilde{\chi}^{MF^{-1}}\big{)}_{xy}=\frac{1}{\beta}\frac{1}{1-m_{B_{x}}^{2}}-N_{x}\Omega_{xy}.$
(40)

We can identify the magnetisation gradient at the level of blocks by solving:

$\tilde{S}\nabla_{\tilde{\underline{h}}_{B}}M^{MF}=\mathbf{1}.$ (41)

###  1.3 Monte Carlo simulations

The standard method to obtain samples from intractable probability
distributions, such as $\mathbb{P}(\underline{s})$ is via Monte Carlo
simulations newman1999monte . The _Markov chain Monte Carlo_ method involves
drawing a series of samples
$\underline{s}_{1},\underline{s}_{2},\underline{s}_{3},...$ from
$\mathbb{P}(\underline{s})$ by constructing a Markov chain which has
$\mathbb{P}(\underline{s})$ as a stationary distribution.

Given some current state of the system $\underline{s}$ there are many ways to
choose a next state $\underline{s}^{\prime}$. A simple choice is _single spin
flip dynamics_ for which we select a spin at random to flip at each timestep.
We use the _Methopolis Hastings_ acceptance probability, for which the
probability of transitioning from $\underline{s}$ to $\underline{s}^{\prime}$
at a given time step in the dynamics is given by:

$A(\underline{s}\rightarrow\underline{s}^{\prime})=\begin{cases}e^{-\beta\Delta
E}\quad\mathrm{if}\quad\Delta E>0\\\ 1\quad\mathrm{otherwise}\end{cases},$
(42)

where the change in energy is given by:

$\Delta E=2s_{i}\bigg{(}\sum_{j\in\mathcal{N}(i)}s_{j}+g_{i}\bigg{)},$ (43)

where $\mathcal{N}(i)$ represents the set of neighbours of node $i$. This is a
generalisation of Equation (3.10) from Ref. newman1999monte  which also
includes the external field $g_{i}$. This is the most efficient choice of the
acceptance probability newman1999monte . The intuition behind this choice is
that we accept any spin flip which lowers the energy of the system. For flips
which increase the energy of the system we accept the flip with a probability
which decays exponentially with the increase in energy.

###  1.4 Setting the temperature scale

In the main text we consider influencing of Ising systems both above and below
the phase transition. For the purposes of this paper it is not necessary to
estimate the critical temperature precisely, however, it is relevant to obtain
a guideline value using the mean-field approximation. We estimate this
baseline for the case of zero ambient field.

For the purpose of this analysis we treat the mean-field equations as a
discrete time dynamical system. In the presence of no external fields
($\underline{g}=0$) the mean-field equations take the form:

$m_{i}^{t+1}=\tanh\bigg{(}\beta\big{(}\sum_{j=1}^{N}A_{ij}m_{j}^{t}\big{)}\bigg{)}=\mathcal{F}_{i}(\underline{m}^{t}),$
(44)

where $\mathcal{F}_{i}(\cdot)$ represents the $ith$ component of the map. This
system always has a fixed point at $\underline{m}=0$.

Following Ref. lynn2016maximizing  we identify the critical temperature by
identifying the conditions under which the fixed point at $\underline{m}=0$ is
the only fixed point. The contraction mapping theorem states that a fixed
point is unique if a map is a contraction brooks2009contraction .
Consequently, if we can identify conditions under which Equation 44 is not a
contraction then the fixed point is necessarily not unique.

We can analyse the stability of a fixed point by considering the Jacobian
matrix $J$, which has elements $J_{ik}=\frac{\partial\mathcal{F}_{i}}{\partial
m_{k}}$. For Equation 44 the elements of the Jacobian matrix take the form:

$J_{ik}=\beta\bigg{(}1-\tanh^{2}\bigg{(}\beta\big{(}\sum_{j=1}^{N}A_{ij}m_{j}\big{)}\bigg{)}\bigg{)}A_{ik}=\beta(1-m_{i}^{2})A_{ik}.$
(45)

The eigenvalues of $J$ evaluated at $\underline{m}=0$ take the form
$\beta\lambda_{i}$, where $\lambda_{1}\leq\lambda_{2}\leq...\leq\lambda_{N}$
are the eigenvalues of the $A$. If any of the eigenvalues of $J$ are greater
than one then the fixed point will be unstable along the direction associated
with the corresponding eigenvector. For the fixed point at $\underline{m}=0$
this will be the case if $\beta\lambda_{N}>1$. Increasing $\beta$ sufficiently
to meet this condition will result in a bifurcation. During this bifurcation
the fixed point at $\underline{m}=0$ becomes unstable and we observe the
formation of two new fixed points. If the largest eigenvalue is degenerate
then it is also possible for more than two fixed points to form during the
bifurcation.

We therefore define the _critical temperature_ of the system by:

$\beta_{c}=\frac{1}{\lambda_{N}}.$ (46)

This formula applies when $\underline{g}=0$, however, we find in practice that
$\beta_{c}$ gives a good guideline value for the interface between the high
and low temperature regimes when ambient fields are present.

###  1.5 Parameters used in Figure 1

In order to generate the results for Figure 1 we implement projected gradient
ascent (Algorithm 2) at the level of the full graph and at the level of
blocks. We use step sizes of $\epsilon=50$ and $1$ and tolerances of
$a=10^{-6}$ and $a=10^{-8}$ for the full and block level IIM respectively. In
each instance of the above algorithms we solve for the mean-field
magnetisation using Algorithm 1 with $\gamma=1$ and tolerance of
$\tau=10^{-5}$.

##  2 Information required to implement different influence strategies

Table S1 summarises the information required to implement each of the
influence strategies considered in the main text.

Influence strategy | Information required | Parameters
---|---|---
Full graph IIM |  $A$, $\underline{b}$, $\beta$, $\underline{m}^{0}$ |  $\epsilon$, $a$, $\gamma$, $\tau$
Block level IIM |  $\mathcal{K}$, $\tilde{\underline{b}}$, $\beta$, $\underline{m}_{B}^{0}$ |  $\epsilon$, $a$, $\gamma$, $\tau$
Swing voter | $\tilde{\underline{b}}$ | $\psi$
Negative canceller | $\tilde{\underline{b}}$ | N/A
Survey-snapshot | $\underline{\hat{M}}_{B}$ | $F_{\mathrm{samp}}$

Table S1: Table illustrating the information about the Ising system required
to implement the different influence strategies implemented in the main text.
For the full graph and block level IIM strategies $\underline{m}^{0}$ and
$\underline{m}_{B}^{0}$ can be any initial condition which converges to the
desired metastable solution under the dynamics in Algorithm 1. Setting
$\underline{m}^{0}$ and $\underline{m}_{B}^{0}$ requires knowledge of the
metastable state which the system is assumed to be in. For main text Figures 1
and 3 we select the metastable solutions with the most positive and most
negative magnetisations respectively. These can be reached by initialising at
$(1,1,...,1)$ and $(-1,-1,...,-1)$ respectively. When implementing the Survey-
snapshot strategy in this paper we use Monte Carlo simulations in order to
estimate the magnetisation at the level of blocks $\underline{\hat{M}}_{B}$.
However, in practice we would aim to set $\underline{m}^{0}$,
$\underline{m}_{B}^{0}$ and $\underline{\hat{M}}_{B}$ from empirical survey
data (e.g. as in Refs. godoy-lorite2020inference  and burioni2015enhancing ).

##  3 Initial conditions for computing phase diagrams

Ising systems consisting of weakly connected groups of spins can possess
metastable states where the groups are polarised in a certain direction. For
the three-block system considered in Section II.2 we can imagine $2^{3}=8$
different metastable states occurring. The addition of ambient external fields
will break the symmetry of the system. This will result in states in which the
spins are aligned with the ambient fields occurring with a higher probability.
In order to determine the full phase diagram of the system, we must take these
metastable states into account. We will now describe how we achieve this for
the mean-field approximation and the Monte Carlo simulations.

Initialisation of block level fixed point iteration.  For the block level
approximation on the three-block system described above
$\underline{m}\in[1,1]^{3}$. Initialising Algorithm 1 at different initial
states when computing the mean-field magnetisation will result in convergence
to different solutions. As noted previously in Section 1.1, it is not easy to
determine the basins of attraction for the convergence of the Algorithm 1.
Consequently, we identify different solutions by sampling _uniform_ random
initial conditions within the domain. In order for this approach to work, we
have assumed that the basins of attraction for the different metastable states
are sufficiently large that we can reach them after a moderate number of
random samples. We use 100 random initial conditions for each value of $\beta$
when making Figure 2b.

Initialisation of full graph fixed point iteration.  When solving the mean-
field self-consistency equations for the full graph we find that using
uniformly random initial conditions in $[-1,1]^{N}$ identifies relatively few
metastable states. This may be because certain metastable states, such as
those close to being fully aligned, have basins of attraction in the far
corners of the high-dimensional hypercube $[-1,1]^{N}$. Consequently, we
instead randomly sample initial states of the form:

$\underline{m}_{0}^{\prime}=(a_{1}\mathbf{1},a_{2}\mathbf{1},a_{3}\mathbf{1}),$
(47)

where $a_{i}\sim U[-1,1]$. A particular random draw of $a_{i}$ will set the
magnetisation of all nodes in block $i$ to be $a_{i}$. This sampling strategy
assumes that all the nodes in a given block have approximately the same
average magnetisation which is a reasonable assumption for weakly coupled
modular graphs. We used 16 random initial conditions for each value of $\beta$
when making Figure 2b.

Initialisation of Monte Carlo simulations.  Choosing to initialise Monte Carlo
simulations randomly (i.e. with +1/-1) will lead to an initial state with
$M\approx 0$. Starting from such an initial condition, it is more likely for
the system to come to a temporary equilibrium in a more disordered metastable
state. This makes it difficult to identify the more aligned metastable states
(for values of $\beta$ where they exist). Consequently, rather than waiting a
long time for the system to transition between different metastable states we
instead use an analogue of Equation 47. We will consider an initial condition
of the form:

$\underline{m}_{\mathrm{MC}}=(\underline{\eta}(a_{1}),\underline{\eta}(a_{2}),\underline{\eta}(a_{3})),$
(48)

where $\underline{\eta}(a_{i})$ is a vector of length $N_{i}$ for which
$\eta(a_{i})_{i}=+1$ with probability $a_{i}$ and $\eta(a_{i})_{i}=-1$ with
probability $1-a_{i}$.

In our simulations we combine samples from multiple Monte Carlo chains after
burn in. We then bin the sampled magnetisation values into equal sized bins in
$[-1,1]$. This allows us to build up a picture of the distribution of
magnetisations for each value of $\beta$.

The solution to the full graph mean-field approximation shown in Figure 2b
lacks symmetry at the first bifurcation point. We have not fully investigated
the cause of this. We expect that the particular SBM draw has an imbalance in
the number of connections between the edge blocks and the centre block. This
will mean that one of the two metastable states will become stable marginally
earlier than the other. We expect that it would also be possible to identify
this using Monte Carlo simulations, however, it is difficult to see this
effect due to noise given the system size considered. We expect such
asymmetries in the solution to the full graph mean-field approximation might
become diminished for larger system sizes.

##  4 Data processing and simulation methodology associated with the Pokec
social network

###  4.1 Pokec Dataset

In order to demonstrate the applicability of the algorithm we will consider
the IIM problem on a real world social network. We use a dataset obtained from
the online social network Pokec takac2012data . This dataset was access from:
https://snap.stanford.edu/data/soc-Pokec.html. The Pokec network contains
$N=1,632,803$ attributed nodes. The network data is directed but we will treat
the edges as undirected.

The 58 different attributes consist of a range of numerical and text fields, a
significant fraction of which are incomplete. We will focus on age and
location as these attributes are relatively well populated compared to the
other attributes. Furthermore, we also expect to see homophily with respect to
these two attributes meaning that we can classify nodes into blocks without
having to rely on knowledge of the graph structure in order to determine the
communities.

Obtaining spatial coordinates.  The locations of users are not stored as
geographic coordinates. Instead, we have the names of the region or towns that
the users are based at. These are in Slovak and stored in the column named
‘region’. In order to convert the region names to geographic coordinates we
use one of the geocoders from the geopy package (see:
https://pypi.org/project/geopy/). This allows us to extract the latitude
longitude coordinates for all of the place names. A small percentage of the
queries fail due to unrecognised place names. In addition, there also exist
users with the term ‘zahranicie’ which translates to abroad. We recover
approximate latitude longitude coordinates associated with these users,
however, we will not consider these profiles in the analysis.

After the above data processing we are left with 188 unique location names.
After keeping valid latitude longitude coordinates and ages we are left with
64% ($N=1042110$) of the original nodes.

###  4.2 Coarse graining using node demographics

We will consider a smaller subgraph of Pokec with supposed block structure. We
consider a subset of the regions around Bratislava in the Pokec dataset as
this domain contains several regions with a large population which are
relatively well connected. Starting from the set of valid profiles described
above we extract a large connected subgraph from the data as follows:

  1. 1.

Select the set of profiles with region names containing the string
‘bratislava’.

  2. 2.

We then select the 3 regions with the largest population:

    1. (a)

‘bratislavsky kraj, bratislava - petrzalka’

    2. (b)

‘bratislavsky kraj, bratislava - ruzinov’

    3. (c)

‘bratislavsky kraj, bratislava - okolie’

Petrzalka and Ruzinov correspond to two densely populated regions on the right
hand side and left hand side of the river, while ‘Okolie’ refers to the
surrounding districts.

  3. 3.

We take the subgraph of users within these regions and then extract the LCC.
The motivation for focusing on the LCC is that disconnected nodes will be
unaffected by the optimisation on the graph structure.

  4. 4.

The age distribution of Pokec users is skewed towards relatively young users.
The 10th and 90th percentiles lie at ages 14 and 37 respectively. In order to
obtain approximately equal sized age groups we will rely on percentiles rather
than age categories of equal size. We consider age bins with boundaries at the
25th,50th and 75th percentiles. Using the age distribution of all users
containing ages this corresponds the bins:

    1. (a)

1-17

    2. (b)

18-21

    3. (c)

22-28

    4. (d)

29-112

Some of the larger ages in the dataset appear to be incorrectly reported.
However, this only impacts a small number of the nodes so will not
substantially skew the results. The resulting connected undirected subgraph
from Pokec contains $N=29582$ users divided into 12 different blocks. The
sizes of the different blocks are reported in table S2. Counting the edges
between the blocks and inverting Equations 4 and 3 allows us to obtain the
coupling matrix, $\mathcal{K}$, for the system. Figure 3a shows a heatmap of
the coupling strengths between the different blocks extracted. The diagonal
element is largest indicating the presence of attribute homophily. We also
observe larger off-diagonal elements which indicate the presence of homophily
between age categories in different regions. We also visualise the coupling
between the different groups in Figure S1a.

###  4.3 Ambient field

Let $\mathcal{A}_{i}$ be the average age of the individuals in block $i$ (we
use the average of the original age bins before diving into regions for
simplicity). We first standardise the ages so that they have a mean of zero
and lie in the range $[-1,1]$. Let:

$\hat{\mathcal{A}}_{i}=\frac{\mathcal{A}_{i}-\langle\mathcal{A}_{i}\rangle}{\max\mathcal{A}_{i}-\min\mathcal{A}_{i}},$
(49)

where $\langle\mathcal{A}_{i}\rangle$ is the average age of block $i$. We then
an consider ambient field on block $i$ be given by:

$\mathcal{B}_{\mathrm{age},i}=g\hat{\mathcal{A}}_{i},$ (50)

where $g$ is a parameter controlling the field gradient. Ambient fields for
the Pokec dataset for $g=10$ are shown in Figure S1b.

Block | Size
---|---
Okolie ages 1-17 | 1234
Okolie ages 18-21 | 1939
Okolie ages 22-28 | 3154
Okolie ages 29-112 | 3458
Petrzalka ages 1-17 | 1398
Petrzalka ages 18-21 | 2106
Petrzalka ages 22-28 | 5234
Petrzalka ages 29-112 | 4550
Ruzinov ages 1-17 | 586
Ruzinov ages 18-21 | 1123
Ruzinov ages 22-28 | 2444
Ruzinov ages 29-112 | 2356

Table S2: Number of nodes within each of the blocks defined based on
individual’s ages and regions.

(a)

(b)

Figure S1: Homophily and ambient field set up in the coarse grained network.
(a) Network plot illustrating the connectivity between different groups in the
Pokec social network (P = ‘Petrzalka’, R=‘Ruzinov’ & O =‘Okolie’). Edge widths
are proportional to the elements of the courpling matrix $\mathcal{K}$ (see
Figure 3a). Nodes are coloured according to the age category (blue: 1-17,
orange: 18-21, green: 22-28 and turquoise: 29-112) and have sizes proportional
to the size of each group (see Table S2). (b) Linear ambient field applied to
the blocks for the case of $g=10$. The three blocks with lower ages have a
small negative field, while the final has a strong positive ambient field.

###  4.4 Phase diagram

Before comparing influence strategies on the Pokec social network it is useful
to explore the circumstances in which the coarse grained approximation of the
network allows us to accurately estimate the magnetisation of the system. We
computed the phase diagram of the system for three different values of the
field gradient $g$. This was performed using Monte Carlo simulations, the full
mean-field approximation and the block level mean field approximation. In the
first two cases we initialise the system only at the fully negative
(-1,-1,…,-1) and fully aligned (1,1,…,1) with the goal of identifying the most
positive and most negative metastable states. In the case of the block level
approximation we sample different random initial conditions for each value of
$\beta$ in a manner comparable to that described in section 3.

Figure S2 shows the phase diagram of the system for three different values of
the field gradient $g$. The block level system was found to have a numerous
metastable states at larger values of $\beta$. The different metastable
solutions correspond to solutions with different blocks being aligned and
anti-aligned with each other. These solutions will not exactly match those in
the full system as the partitions are relatively coarse and based on
attributes rather than being inferred from any observed modular structure in
the network. Nonetheless, the behaviour of the average magnetisation as a
function of $\beta$ is broadly consistent with that obtained using the full
graph mean-field approximation and Monte Carlo simulations. As the field
gradient is increased we observe that some of the positive metastable
solutions become suppressed and exist for a smaller range of $\beta$ values.
This results from 3 out of 4 of the age groups having a negative ambient field
(see Figure S1b).

(a)

(b)

(c)

Figure S2: The magnetisation of the Pokec social network can be estimated
using a coarse grained approximation. Phase diagram for the Pokec social
network computed using Monte Carlo simulations (green circles), mean-field
approximation on the full graph (blue squares) and the mean-field
approximation on the coarse grained system (red triangles). Shown for
different values of the linear field gradient (a) $g=1$, (b) $g=5$ and (c)
$g=10$. For the full graph mean-field approximation and Monte Carlo
simulations we only consider the most positive and negative solution branches.
Different metastable solutions were obtained for the block level approximation
by sampling 100 random initial conditions in the fixed point iteration
(Algorithm 1) for each value of $\beta$. Fixed point iteration was carried out
with $\gamma=1$, $\tau=10^{-5}$ and a max number of iterations of $10^{4}$ for
both the full graph and block level approximation. For each value of $\beta$
we evaluate the average magnetisation using a single Monte Carlo chain with
$T=10^{4}$ and a burn in time of $T_{\mathrm{burn}}=3\times 10^{5}$. We
perform the simulations for 25 equally spaced (on a log base 10 scale) values
of $\beta/\beta_{c}$ varying between $10^{-1}$ and $10^{1.8}$, where
$\beta_{c}$ is obtained from the spectral radius of the adjacency matrix of
the full graph.

###  4.5 Impact of sample fraction on survey-snapshot control

The survey-snapshot control introduced in Section II.2.1 is derived from a
snapshot of the Ising system’s state at a particular point in time. The
fraction of nodes sampled within each block will impact how well we can
estimate the magnetisation of the system and therefore how well we can
influence the system.

For a particular snapshot, we can quantify the quality of our estimate of the
average magnetisation of a block $B_{i}$ for a particular value of
$F_{\mathrm{samp}}$ (the fraction of nodes sampled) by computing the absolute
deviation from the average magnetisation. We can quantify this deviation with
the quantity:

$\delta\hat{M}_{B_{x}}(F_{\mathrm{samp}})=|\hat{M}_{B_{x}}(F_{\mathrm{samp}})-\langle\hat{M}_{B_{x}}(1)\rangle|,$
(51)

where $\langle\cdot\rangle$ indicates the average over multiple samples.
Averaging this quantity over blocks allows us to obtain an overall picture of
how accurately we can estimate the magnetisation for a particular fraction of
nodes sampled. There are two sources of variance in
$\hat{M}_{B_{x}}(F_{\mathrm{samp}})$: that due to the sub-sample of nodes and
that due to the particular snapshot in time. Averaging
$\delta\hat{M}_{B_{x}}(F_{\mathrm{samp}})$ over multiple draws allows us to
assess the impact of the former.

Figure S3a shows the behaviour of $\langle\delta\hat{M}_{B_{x}}\rangle$ as a
function of $F_{\mathrm{samp}}$ for 4 of the blocks in the system. We also
show the behaviour of
$\delta\hat{M}=\frac{1}{q}\sum_{x=1}^{q}\langle\delta\hat{M}_{B_{x}}\rangle$,
which represents the average of the absolute deviation across all the blocks.
We observe that error shrinks relatively rapidly for values of
$F_{\mathrm{samp}}$ between $10^{-3}$ and $10^{-2}$.

In order to explore the performance of the survey-snapshot control as we
increase the sample size we draw a number of survey-snapshots and evaluate the
survey-snapshot control for each of these. Figure S3b illustrates how the
consistency of the value of $M(\underline{h})$ obtained for different
snapshots increases as a function of $F_{\mathrm{samp}}$. The average
magnetisation across all instances of the survey-snapshot strategy remains
stable with sample size. However, the variance in performance of the
strategies decreases as $F_{\mathrm{samp}}$ increases indicating that we can
achieve more consistent results by sampling more of the population.

For small values of $F_{\mathrm{samp}}$ we may sample single nodes for each
block or a small number of nodes in the same state. In this case the estimated
magnetisation for a block will be $\pm 1$. When this is the case the quantity
used to capture the susceptibility $\beta(1-m_{i}^{2})$ will be equal to zero.
When $F_{\mathrm{samp}}$ is sufficiently small for this to occur for all
blocks, all of the elements of the susceptibility vector will be zero meaning
that we cannot meaningfully assign a control field using our approximation.
Consequently, we do not consider values of $F_{\mathrm{samp}}$ significantly
lower than $10^{-3}$.

(a)

(b)

Figure S3: The average performance of the survey-snapshot influence strategy
is unaffected by sample size, however, consistency increases with fraction of
nodes sampled. (a) Plot showing how $\langle\delta\hat{M}_{B_{i}}\rangle$
(averaged over 10 realisations of $\hat{M}_{B_{i}}(F_{\mathrm{samp}})$) varies
as a function of $F_{\mathrm{samp}}$ for 4 of the blocks in the system. The
true magnetisation of each block $\langle\hat{M}_{B_{i}}(1)\rangle$ is
estimated from the average of 10 Monte Carlo simulations. We also show the
behaviour of the average deviation across all blocks $\delta\hat{M}$ (black
line). (b) Plot showing $M(\underline{h})$ for different instances of
$\underline{h}$ obtained using the survey-snapshot influence strategy (see
main text Section II.2.1) as a function of $F_{\mathrm{samp}}$. Scatter points
show average mangetisations obtained using 15 different instances of the
survey-snapshot control. Black squares show the average value across these
instances with the error bars indicating the standard error on the mean.
Average mangetisations are computed using the average of 10 Monte Carlo chains
with a burn in time of $3\times 10^{5}$ and tun time of $10^{4}$. All
simulations performed for the case of $g=1$, $\beta=8\beta_{c}$ with the
system initialised at (-1,-1,…,-1) on the Pokec social network.

##  5 Details of Code used for Numerical simulations

The code used to perform simulations and generate the plots is available at:
https://github.com/MGarrod1/unobserved_spin_influence

This source code relies on two modules created by the author:

  * •

https://github.com/MGarrod1/ising_block_level_influence

  * •

https://github.com/MGarrod1/spatial_spin_monte_carlo

The first is used to compute the magnetisation of Ising systems on networks
under the mean-field approximation and the second is used to carry out the
Monte Carlo simulations in order to validate the algorithms.

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